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Compliments  of  the 

U.  S,  HYDROGRAPHIC  OFFICE, 

COMMANDER  J.  R.  BARTLETT,  Q.  S.  N., 

Hydrograpfier. 

Washington,  D.  C. 


FINDING   THE   ERROR 


OP 


THE   MARINE  COMPASS 


OIST    BOA.RD    SHIP. 


BY 


B.  F.  GREENE, 

PROFESSOR  OF  MATHEMATICS,   UNITED   STATES   NAVY, 
SUPERINTENDENT   OF   COMPASSES. 


OF 
NAVY   DEPARTMENT. 


WASHINGTON: 

GOVERNMENT    FEINTING     OFFICE. 

1875. 


PREFATORY  NOTE. 


So  much  of  tiie  Text,  comprising  nearly  all  of  one  Chapter,  together 
with  the  entire  set  of  Tables,  as  here  given,  from  the  Manual  of  the 
Marine  Compass,  in  course  of  preparation  by  the  undersigned,  it  has 
been  deemed  expedient  to  print,  in  advance  of  its  general  publication, 
for  the  immediate  use  of  the  United  States  Navy. 

Leaving  any  statement  of  particulars,  as  to  the  author's  intentions, 
peculiarities  of  treatment,  and  acknowledgments,  to  a  more  appropriate 
time,  he  would  merely  remark,  at  present,  that  his  aim  has  been,  so  far 
as  "Finding  the  Error  of  the  Compass"  is  concerned,  to  aid  the  Navi- 
gator with  some  additions  to  his  appliances  for  an  easy,  expeditious, 
and  reliable  determination  of  this  important  element.  And  this  is 
done,  from  the  point  of  view  occupied  by  him,  That,  because  there  is  so 
much  unavoidable  liability  to  error  and  consequent  uncertainty  in  the  use 
of  the  Compass  at  sea,  there  is  the  greater  reason  why  ice  should  neglect  no 
available  means  for  reducing  these  errors  on  the  one  hand  and  insuring 
greater  certainty  on  the  other.  If  he  shall  succeed  in  commending  this 
idea  to  the  approval  of  Navigators,  arid  in  making  their  arduous  duties 
enough  easier  in  this  respect  to  induce  a  more  frequent  determination 
of  their  Compass-Errors,  he  will  feel  somewhat  compensated  for  the 
labor  bestowed  even  upon  this  one  Chapter  of  his  work,  not  to  mention 
that  involved  in  the  preparation  of  the  appended  Tables. 

B.  P.  GKEENE. 


BUREAU  OF  NAVIGATION,  NAVY  DEPARTMENT, 
Washington,  March,  1875. 


CONTENTS. 


§  1. —NATURE   OF   THE   COMPASS-ERROR. 

Art.  Page. 

1.  Different  Errors  of  the  Compass 1 

2.  Pointing-Errors  of  the  Compass 1 

a)  Adjustment- Errors  of  the  Compass. 

b)  Adjustment- Err  or  8  of  the  Azimuth-Circle. 

c)  Error  from  Defective  Sensibility. 

d)  Observation- Errors  of  the  Compass. 

3.  Reduction-Errors  of  the  Compass 3 

a)  The  Compass-Deviation. 

b)  The  Magnetic  Variation. 

4.  Total  Error :  called  simply  the  Compass-Error 3 

Two  Different  Definitions  of  the  Compass-Error. 

§  2. — FINDING  THE   COMPASS-ERROR. 

5.  The  Basis  of  Procedure  in  all'  Cases  4 

6.  Different  Methods  of  Finding  the  True  Azimuth •        4 

I.  BY  OBSERVATIONS  OF  CELESTIAL  OBJECTS. 

A.  METHOD  OF  HORIZON-AZIMUTHS. 

7.  Fundamental  Principles  of  the  Horizon- Azimuth 6 

8.  Remark  :  The  Amplitude 6 

9.  Rule  :  To  make  the  Observation 7 

a)  Observation  of  the  Sun. 
&)   Observation  of  the  Moon. 

c)  Observation  of  a  Planet  or  Fixed  Star. 

10.  Remark:  Auxiliary  Observations 7 

11.  Rule  :  To  correct  the  Observed  Compass- Azimuth 8 

«)  For  the  Sun,  a  Planet,  or  Fixed  Star. 

b)  For  the  Moon. 

12.  Examples  of  Correcting  Observed  Azimuths 8- 

13.  Rule  :  To  find  the  True  Azimuth 8 

a)  Preparation  of  the  Data. 

b)  Solution  by  Computation. 

c)  Examples  of  True  Azimuths  by  Computation. 

d)  Solution  by  Inspection  :   Use  of  Tab.  XXIV. 

e)  Remark :  On  taking  Means  of  Tabular  Quantities. 

f)  Examples  of  True  Azimuths  by  Inspection. 

14.  Examples  of  Finding  the^Cornpass-Error  by  Horizon-Azimuths 11 

B.  METHOD  OF  TIME-AZIMUTHS. 

15.  Fundamental  Principles  of  the  Time- Azimuth 12 

16.  The  Ship-Time  ;  its  Importance,  etc 12 


VI  CONTENTS. 


Art.  I'ajie. 

17.  Finding  the  Compass-Error  for  a  Single  Heading  of  the 

Ship 13 

18.  To  make  a  Single  Time-Azimuth  Observation 13 

19.  Finding  the  True  Azimuth 14 

20.  Rule  :  Preparation  of  the  Data 14 

a)  The  Object  being  the  Sun. 

&)  The  Object  being  the  Moon,  a  Planet,  or  Fixed  Star. 

Remark  on  taking  out  the  R.  A. 

21.  Examples  in  Finding  the  Hour- Angle 16 

22.  Rule  :  Solution  by  Logarithmic  Computation 17 

23.  Examples  of  True  Azimuths  by  Computation 17 

24.  The  Tables  of  Time-Azimuths 18 

a)  Tab.  XXX,  or  First  Part. 

6)  Tab.  XXX.  A,  or  Second  Part. 

c)  Tab.  XXX.  B,  or  Third  Part. 

d)  Intention,  etc.,  u'ith  respect  to  these  Tables. 

25.  Direct  and  Limiting  Values  of  Time- Azimuths 20 

26.  Rule  :  Solution  by  the  Azimuth-Tables 20 

27.  Examples  of  True  Azimuths  by  the  Azimuth-Tables '21 

28.  Remark  I :  On  the  Two  Cases  of  the  Time-Azimuth 22 

29.  Remark  II :  On  the  Conditions  introducing  the  Use  of  the  Symbols  — oo  and 

-{- oc , 23 

a)  The  Dec.  equal  to  the  Lat. 

6)  The  Dec.  nearly  equal  to  the  Lat. 

30.  Examples  of  Finding  the  Compass-Error  by  Time-Azimuths 24 

31.  Finding    the  Compass-Error    f o  •    a    Series  of  Different 

Headings  of  the  Ship 25 

32.  To  make  the  Observations  of  Serial  Time-Azimuths 25 

Remark  I :  Position  of  the  Object. 

Remark  IT:  Steadying  the  Ship. 

Remark  III:  Heading- Intervals. 

Remark  IF:  Care  in  Sighting  and  Reading  off. 

33.  Rule :  To  get  the  Serial  True  Azimuths  by  Computation 26 

34.  Rule  :  To  get  the  Serial  True  Azimuths  by  the  Azimuth-Tables 27 

35.  Examples  of  Finding  Serial  Compass-Errors  by  Time- Azimuths 28 

C.  METHOD  OF  CIRCUMPOLAR  AZIMUTHS. 

36.  The  Circumpolar  Azimuth  a  Modified  Form  of  the  Time- Azimuth 32 

37.  Table  of  Circumpolar  Azimuths  for  Polaris  :  Description  of  Tab.  XL 32 

38.  Two  Distinct  Problems  in  the  Use  of  Tab.  XL 33 

39.  Rule  :  To  find  the  True  Azimuth  of  Polaris 33 

a)  To  take  out  the  True  Azimuth  for  any  Ship-Time. 

6)  To  find  the  Ship-Time  of  the  greatest  W.  or  E.  E.  of  Polaris. 

40.  Examples  of  Finding  the  Compass  Error  by  Circumpolar  Azimuths 34 


D.  METHOD  OF  ALTITUDE-AZIMUTHS. 

41.  Fundamental  principles  of  the  Altitude-Azimuth 35 

42.  Rule  :  To  make  the  Observation 35 

43.  Rule :  To  find  the  True  Azimuth  by  Computation 35 

44.  Rule :  To  find  the  True  Azimuth  by  the  Azimuth-Tables 36 

Remark  I :  Arguments  to  Odd  Numbers. 

Remark  II:  Direct  and  Limiting  Values  of  Altitude- Azimuths. 

45.  Examples  of  Finding  the  Compass-Error  by  Altitude-Azimuths 37 


CON-TENTS.  VII 


E.  METHOD  OF  TIME- ALTITUDE-AZIMUTHS. 

Art.  Page. 

46.  Fundamental  Characteristics ;  Advantages  and  Defects 38 

47.  Rale  :  To  find  the  True  Azimuth  by  Computation 38 

48.  Examples  of  Finding  the  True  Azimuth 39 

49.  Table  of  Time-Alt.  Azimuths :  Use  of  Tab.  XLVIII 39 

50.  Examples  of  the  Use  of  Tab.  XLVIII 40 

F.  METHOD  OF  TRANSITION- AZIMUTHS. 

51.  Fundamental  Principles  of  the  Transition-Azimuth 40 

52.  Remark  on  the  Preparation  of  the  Data 41 

53.  Rule:  To  find  the  True  Azimuth 41 

«)  By  Logarithmic  Computation. 
•  &)  By  Tabular  Inspection. 

54.  Examples  of  Finding  the  Compass-Error  by  Transition- Azimuths 42 

G.  DEPENDENCE  TO  BE  PLACED  ON  THESE  METHODS. 

55.  The  Data  of  a  True  Azimuth  always  liable  to  be  in  Error 42 

56.  Estimating  Errors  of  the  Azimuth-Data 42 

a)  The  Latitude-Error. 
6)  The  Declination-Error. 

c)  The  Hour-Angle  Error. 

d)  The  Altitude-Error. 

e)  Recapitulation  of  the  Limits  of  the  Data-Errors. 

57.  Tables  of  Azimuth-Errors  :  Auxiliary  Tables 44 

a)  For  Errors  of  Horizon- Azimuths. 
&)  For  Errors  of  Time- Azimuths. 

c)  For  Errors  of  Altitude- Azimuths. 

d)  For  Errors  of  Time-Alt.  Azimuths. 

58.  Finding  Partial  Azimuth-Errors 45 

a)  Partial  Errors  of  Horizon- Azimuths. 
I)  Partial  Errors  of  Time- Azimuths. 

c)  Partial  Errors  of  Altitude- Azimuths. 

d)  Partial  Errors  of  Time-Alt.  Azimuths. 

59.  Probable  Total  Azimuth-Error 47 

60.  Favorable  and  Unfavorable  Conditions 48 

a)  Conditions  of  Horizon-Azimuths. 

b)  Conditions  of  Time- Azimuths. 

c)  Conditions  of  Altitude- Azimuths. 

d)  Conditions  of  Time-Alt.  Azimuths. 

61.  Limits  of  allowable  Azimuth  Error 49 

62.  Dependence  on  Horizon-Azimuths 51 

63.  Dependence  on  Time-Azimuths 52 

a)  Single  Time- Azimuths. 

b)  Serial  Time- Azimuths. 

64.  Dependence  on  Circumpolar  Azimuths 54 

65.  Dependence  on  Altitude- Azimuths 54 

a)  Single  Altitude- Azimuths. 

b)  Serial  Altitude- Azimuths. 

66.  Relative  Advantages  of  these  Methods 56 

a)  Horizon-Azimuths. 

b)  Time-Azimuths. 

c)  Circumpolar  Azimuths. 

d)  Altitude- Azimuths. 


VIII  CONTENTS. 


II.  BY  OBSERVATIONS  OF  TERRESTRIAL  OBJECTS. 
A.  METHOD  BY  DIRECT  BEARINGS. 

Art.  Page- 

07.  Fundamental  Conditions 58 

o)  First  Condition  :  A   Fixed  Station. 

&)  Second  Condition:'  Means  of  Swinging  the  Ship. 

c)  Third  Condition  :  A  Limited  Parallax  of  Swing. 

68.  Process  of  Finding  Serial  Compass-Errors 60 

69.  Examples  of  the  Method  by  Direct  Bearings 61 

70.  Dependence  to  be  placed  on  Results <>:> 

71.  Tables  of  Parallactic  Errors  :  Examples ^3 

72.  To  correct  the  Observations  for  Parallactic  Errors 6:5 

a)  Case  of  Using  the  True  Bearing  of  Object. 
&)  Case  of  Using  the  Magnetic  Bearing. 

c)  Remark  I:  Observing  two  or  more  Objects. 

d)  Remark  II:  Swinging  with  a  Taut  Cable. 

e)  Remark  III :   Using  the  nearest  whole  Degree. 

73.  Examples  of  Correcting  for  Parallactic  Errors 65 

74.  Limiting  Distance-Ratios 67 

75.  Different  Methods  of  Finding  the  True  Bearing  of  Object 67 

76.  Finding  True  Bearing  by  the  Geographical  Position 67 

77.  Examples  of  Finding  T.  B.  .by  the  Geographical  Position 68 

78.  Finding  True  Bearing  by  an  Astronomical  Bearing : firt 

a)  Rule :  To  make  the  Observation. 
6)  Rule :  To  make  the  Computation. 

79.  Examples  of  Finding  T.  B.  by  an  Astronomical  Bearing 69 

80.  Finding  True  Bearing  by  an  Azimuth  of  the  Vertical  Circle 70 

a)  Rule:  To  take  the  Observation. 

6)  Rule:  To  find  the  True  Bearing  of  the  Object. 

81.  Examples  of  Fuding  T.  B.  by  an  Azimuth  of  the  Vertical  Circle 71 

82.  Remarks  relative  to  Finding  the  True  Bearing  of  the  Object 71 

B.  METHOD  BY  ALLIGNMENTS. 

83.  Fundamental  Idea 72 

84.  Process  of  Finding  Serial  Compass-Errors 72 

85.  Examples  of  the  Method  by  Alignments 73 

86.  Dependence  to  be  placed  on  Results 74 

C.  METHOD  BY  RECIPROCAL  BEARINGS. 

87.  Preliminary  Explanat ions 74 

88.  Two  Distinct  Cases  of  Reciprocal  Bearings 75 

89.  Essential  Kequisites  of  the  Method  under  the  First  Case 75 

90.  Essential  Requisites  of  the  Method  under  the  Second  Case 76 

a)  The  Angles  measured  from  the  True  Meridian. 

b)  The  Angles  measured  from  an  Arbitrary  Line. 

91.  Preparations  for  Observations  by  this  Method % 77 

92.  Process  of  Conducting  Reciprocal  Observations 77 

93.  Reduction  of  Reciprocal  Observations 78 

a)  Magnetic  Observations  on  Shore. 

&)  Angles  Measured  from  the  True  Meridian  on  Shore. 

c)  Angles  Measured  from  any  Arbitrary  Line. 


CONTENTS.  IX 


Art.  Page. 

94.  Examples  of  the  Method  hy  Eeciprocal  Bearings , 79 

95.  Dependence  to  he  placed  on  Results 82 

96.  To  find  the  True  Meridian 82 

97.  Remark :  To  set  the  Instrument  in  a  True  Meridian 83 

98.  Example  of  Finding  the  True  Meridian 83 

D.    RELATIVE    ADVANTAGES    OF    THE     SEVERAL    METHODS     OF     FINDING    SERIAL 

COMPASS-ERRORS. 

99.  Methods  hy  Celestial  Azimuths 84 

100.  Method  hy  Direct  Bearings „ 85 

101.  Method  hy  Reciprocal  Bearings 85 

APPENDIX. 

COMPASS-COMPARISONS. 

102.  Definition 87 

A.  COMPASS-COMPARISONS  ON  BOARD  SHIP. 

103.  Characteristics  and  Uses  of  Comparisons 87 

104.  To  make  a  Compass-Corn  parison 87 

105.  Examples  of  Compass-Comparisons 88 

106.  To  convert  a  Given   Course  hy  Standard  Compass    into  an    Equivalent 

Course  hy'a  Compared  Compass  and  Reciprocally 88 

107.  Examples  of  Direct  and  Reciprocal  Conversions - >. 88 

108.  Rule :  To  find  the  Error  or  Deviation  of  a  Compared  Compass 88 

109.  Examples  of  Fiodjng  the  Errors  of  Compared  Compasses •. 89 

B.  COMPASS-COMPARISONS  ON  SHORE. 

110.  Comparison  on  Shore:  Detection  of  Local  Magnetism 89 


IT 


INDEX  TO  THE  TABLES. 


Table.  Page. 

I.  Compass-Points  and  their  Equivalents  in  Degrees 1 

II.  Conversion  of  Arc  into  Time  and  Time  into  Arc 2 

III.  Conversion  of  Mean  Solar  into  Sidereal  Time 

IV.  Conversion  of  Sidereal  into  Mean  Solar  Time 3 

V.  Length  of  a  Degree  in  Latitude  or  Longitude 4 

VI.  Logarithms  of  Numbers  and  Small  Arcs 5 

VII;  Logarithms  of  Numbers  1  to  1009 24 

VIII.  Logarithmic  Sines,  Tangents,  and  Secants  to  every  Eighth  of  a  Com- 
pass-Point    26 

IX.  Logarithmic  Sines,  Tangents,  and  Secants  to  every  Minute  of  Arc 

and  Fourth  Minute  of  Time 27 

X.  Logarithmic  Sines  to  every  Tenth  of  a  Degree 72 

XI.  Logarithmic  Secants  to  every  Tenth  of  a  Degree 74 

XII.  Logarithmic  Tangents  to  every  Tenth  of  a  Degree  , 76 

XIII.  Logarithmic  Tangents  to  every  Minute  of  Time 78 

XIV.  Lengths  of  Circular  Arcs  to  every  Tenth  of  a  Degree 79 

XV.  Natural  Sines  and  Cosines  to  every  Tenth  of  a  Degree HO 

XVI.  Natural  Tangents  to  every  Tenth  of  a  Degree 82 

XVII.  Natural  Versed  Sines  to  every  Tenth  of  a  Degree 84 

XVIII.  Decimal  Parts  and  their  Multiples  of  a  Day 85 

XIX.  Decimal  Equivalents  of  Common  Fractions 86 

XX.  Proportional  Parts 89 

XXI.  Squares  of  Numbers  from  0.0  to  100.9 90 

XXII.  Square  Roots  of  Numbers  from  0.0  to  100.9 - 92 

XXIII.  True  Rising  and  Setting 94 

XXIV.  Horizon- Azimuths 99 

XXV.  Position-Angles  for  Horizon-Azimuths 104 

XXVI.  Limiting-Errors  of  Horizon-Azimuths 104 

XXVII.  Correction  of  the  Compass- Azimuth  on  the  Apparent  Horizon 105 

XXVIII.  Error  of  the  Horizon-Azimuth  for  an  Error  of  0°.2  in  the  Latitude. ..  105 
XXIX.  Error  of  the  Horizon-Azimuth  for  an  Error  of  0°.l  in  the  Declina- 
tion   105 

XXX.  Time- Azimuths :  Logs  A  and  B 106 

XXX.  A.  Time- Azimuths :  Log  C 124 

XXX.  B.  Time- Azimuths :  Log  Tangents  X  and  Y 130 

XXXI.  Time-Azimuths :  Direct  and  Limiting  Values 132 

XXXII.  Position- Angles  of  Direct  and  Limiting  Time- Azimuths 134 

XXXIII.  Altitudes  of  Direct  and  Limiting  Time- Azimuths '    136 

XXXIV.  Error  of  the  Time-Azimuth  for  an  Error  of  lm  in  the  Hour-Angle 138 

XXXV.  Error  of  the  Time-Azimuth  for  an  Error  of  0°.2  in  the  Latitude 141 

XXXVI.  Error  of  the  Time-Azimuth  for  an  Error  of  0°.l  in  the  Declination. .  -  141 

XXXVII.  Limiting-Errors  of  Time-Azimuths 142 

XXXVIII.  Limiting-Errors  of  Time-Azimuths  in  High  Latitudes 142 

XXXIX.  Limiting-Errors  of  Serial  Time-Azimuths 142 

XL.  Circumpolar  Azimuths  :  Polaris  or  the  Pole-Star , 143 

XLI.  Altitude-Azimuths:  Part  I •. 144 

XLI.  Altitude-Azimuths:  Part  II «. 146 

XLI.  Altitude-Azimuths:  Part  III..                                                   146 


XII  INDEX   TO    THE    TABLES. 


Table.  Page. 

XLII.  Altitude  Azimuths :  Direct  and  Limiting  Values 148 

XLIII.  Position-Angles  of  Direct  and  Limiting  Altitude-Azimuths  , 150 

XLIV.  Error  of  the  Altitude- Azimuth  for  an  Error  of  0°.l  in  the  Altitude. ..  150 
XLV.  Error  of  the  Altitude-Azimuth  for  an  Error  of  0°.2  in  the  Latitude..  151 
XLVI.  Error  of  the  Altitude- Azimuth  for  an  Error  of  0°.l  in  the  Declina- 
tion   152 

XL VII.  Limiting-Errors  of  Altitude-Azimuths 152 

XLVIII.  Time-Alt.  Azimuths :  Log  A 153 

XLVIII.  Time-Alt.  Azimuths  :  Log  B 154 

XLIX.  Error  of  the  Time-Alt.  Azimuth  for  an  Error  of  0»'.2  in  the  Hour- 
Angle  ^ 158 

L.  Error  of  the  Time-Alt.  Azimuth  for  an  Error  of  3  in  the  Altitude  or 

Declination , 158 

LI.  Transition- Azimuths 159 

LII.  Direct  Bearings  of  a  Fixed  Object:  Limiting  Distance 160 

LIII.  Direct  Bearings  of  a  Fixed  Object :  Parallactic  Errors 160 

LI V.  Products  of  Arcs  Multiplied  by  the  Sines  of  the  Rhumbs 161 

LV.  Magnetic  Elements  of  the  Earth:  The  Magnetic  Variation  in  Arctic 

Latitudes 169 

LVI.  Magnetic  Elements  of  the  Earth  :  The  Magnetic  Variation  in  Lati- 
tudes 70°  N  to  60°  S no 

LVII.  Magnetic  Elements  of  the  Earth :  The  Magnetic  Dip 174 

LVIII.  Magnetic  Elements  of  the  Earth  :  The  Horizontal  Force 175 

LIX.  Kight  Ascension  of  the  True  Sun  and  Equation  of  Time 176 

LX.  Declination  of  the  Sun 180 

LXI.  Mean  Places  of  T  wenty-fi  ve  Fixed  Stars 182 

LXII.  Meridian-Passages  of  Twenty-five  Fixed  Stars 182 

LXIII.  Reductions  of  Meridian- Passages  of  the  Fixed  Stars 183 

LXIV.  Reduction  of  Daily  or  Hourly  Changes  in  Right  Ascension 184 

LX V.  Reduction  of  Hourly  Changes  in  the  Moon's  Right  Ascension 186 

LXVI.  Reduction  of  the  Mean  Sun's  Right  Ascension 188 


THE  COMPASS-ERROR,  AND  METHODS  OF  FINDING 

BOARD  SHIP. 


§  1.— NATURE  OF  THE  COMPASS-ERKOR. 

1.  Different  Errors  of  the  Compass.— The  Errors  of  the 
Marine  Compass  may  be  considered  in  two  different  relations:  First,  as 
to  the  pointing  of  the  compass,  which,  but  for  the  influence  of  certain 
errors,  would  always  be  in  precise  accordance  with  the  Directive  Force, 
whether  on  board  ship  or  on  the  land ;  and,  secondly,  as  to  the  reduction 
to  the  True  Meridian  of  the  corrected  compass-pointing,  which,  although 
in  no  proper  sense  an  error  of  the  compass,  is  treated  like  one  from 
considerations  of  practical  convenience.  All  the  errors  of  the  compass 
explained  or  referred  to  in  the  preceding  Chapters1  may  therefore  be 
regarded  as  forming  parts  of  either  the  Pointing-Errors  or  Reduction- 
Errors  of  this  instrument,  and  they  will  now  be  recapitulated  under 
these  two  heads. 

£.  Pointing-Errors  or  the  Compass. — The  Pointing-Errors 
include  all  instrumental  and  observational  errors  which  affect  the  accu- 
racy of  the  compass-pointing.  Under  this  head  there  are  the  following 
errors : 

a]  Adjustment-Errors  of  the  Compass. — These  comprise  (Chap.  I) — 

1.  The  divergence  of  the  zero-line  from  the  magnetic  axis  of  the  com- 
pass-card ; 

2.  The  eccentricity  of  the  cap  with  respect  to  the  centre  of  the  card- 
circle  ;  and 

3.  The  eccentricity  of  the  pivot  with  respect  to  the  centre  of  the 
bowl-circle. 

b)  Adjustment-Errors  of  the  Azimuth- Circle*. — These  are — 

1.  The  Direction-Ernpr  of  the  sight-vanes,  or  divergence  of  the  verti- 
cal plane  of  sight  from  the  diameter  of  the  circle  (Chap.  I) ;  and 

2.  Several  possibilities  of  error  from  defective  construction,  which 
need  not  be  mentioned  in  this  place.     (See  Chap.  I.) 

c]  Error  from  Defective  Sensibility  (Chap.  T). 

d)  Observation- Errors  of  the  Compass. — These  comprise — 

i.  Equilibrium- Errors  of  the  compass-card  and  bowl-circle  (Chap.  V) ; 
and 

1  The  references  here  and  elsewhere  made,  are  to  different  parts  of  the  Manual  of  the 
Marine  Compass,  not  yet  published,  of  which  this  is  one  Chapter  (the  sixth,  with  cer- 
tain modifications),  as  explained  in  the  Prefatory  Note. 


FINDING    THE    COMPASS-ERROR. 


2.  Personal  Errors  iu  sighting  and  reading  off  (Chap.  V). 

The  Adjustment-Errors  of  both  the  compass  and  azimnth-circle 
should  be  practically  unimportant,  when  these  instruments  are  delivered 
by  the  makers;  and  the. adjustments  should  be  sufficiently  stable  to  re- 
main unimpaired  for  a  long  period,  with  careful  and  considerate  usage. 
Unfortunately,  even  the  best  intentions  of  the  most  reliable  maker  are 
liable  to  be  occasionally  frustrated  ;  and  hence  the  only  certain  way  to 
arrive  at  satisfactory  conclusions  concerning  these  errors  is  to  subject 
the  instruments  to  careful  tests  of  their  actual  condition  (Chap.  I). 

The  error  from  Defective  Sensibility  of  a  well-made  liquid  compass, 
having  a  buoyant  card,  adjusted  to  a  minimum  pressure  at  the  pivot,  is 
commonly  inappreciable ;  and,  under  circumstances  ordinarily  favora- 
ble, it  may  in  general  be  relied  on  to  continue  in  that  condition  for  a 
long  period.  But  other  compasses,  whether  air  or  liquid,  with  heavy 
cards,  and  having  a  pivot-pressure  ranging  from  1,500  to  5,000  grains, 
are  liable  to  an  appreciable  error  from  this  source,  proportional,  at  the 
least,  to  the  greater  pressure  at  the  pivot ;  and  not  only  that,  but  such 
compasses  are  even  more  than  proportionally  liable  to  the  development 
of  a  very  serious  error  as  the  cap  and  pivot  become  sensibly  worn  or 
otherwise  altered  in  form  (Chap.  I). 

The  compasses  of  the  United  States  Navy,  under  the  inspection-tests 
to  which  they  are  subjected,  are  required  to  satisfy  the  following 
conditions: 

The  greatest  error  to  which  they  may  be  liable  must  not  exceed — 
For  Adjustment-Errors  of  the  compass  (a)  ±o°.i 
For  Adjustment-Errors  of  the  az.  circle  (I)  rto  .1 
For  'Error  from  Defective  Sensibility  (c)       d=  o  .01 
or,  in  the  last  case,  it  must  be  inappreciable.  Such  a  limitation  of  these 
errors  is  equivalent  to  a  probable  limit,  in  the  aggregate,1  of  ±  o°.i4. 

These  conditions,  or  something  equivalent  to  them,  by  which  th<* 
Navy  compasses  are  required  to  be  "  practically  perfect "  when  put  on 
board  ship,  should  be  required  by  all  Navigators,  or  at  the  least  for  one 
(standard)  compass  of  their  ships. 

The  Observation-Errors,  whether  depending  on  the  circumstances  of 
the  observation  or  on  the  personal  peculiarities  of  the  observer,  are 
alike  impossible  of  previous  estimate.  Nor,  indeed,  can  they  be  esti- 
mated with  much  pretension  to  accuracy — even  as  to  their  limits — after 
the  observation  has  been  made,  unless  conducted'with  a  view  to  such  an 
estimate;  and  this  is  rarely  practicable  or  expedient  with  the  Compass- 
Observations  generally  required  for  the  purpose  of  shaping  or  correct- 
ing a  course.  Nothing  can  therefore  be  relied  on  but  intelligence  and 
care  in  making  these  observations  as  a  remedy  for  this  source  of  Com- 
pass-Error. 

It  is  otherwise  with  observations  of  a  serial  kind,  which  are  made  for 
the  purpose  of  obtaining  a  set  of  Compass-Deviations ;  for  it  is  then 
quite  practicable  to  get  a  definite  estimate  of  the  total  Pointing-Error, 

'The  probable  sum  of  these  errors  is  ±  l/(o°.i)a-f.(o0.i)*  +  (o°.oi)8  =  ±  o  .14. 


NATURE    OF    THE    COMPASS-ERROR. 


including  the  errors  of  observation,  at  least  as  to  its  limits,  both  in  sign 
and  amount  (Chap.  IV). 

3.  Reduction-Errors  of  the   Compass. — The  Eeductiou- 
Errors  include  such  as  are  not,  strictly  speaking,  errors  of  the  compass, 
but  those  which  are  treated  as  if  they  were,  in  the  reduction  of  the  cor- 
rected compass-pointing  to  the  Geographical  or  True  Meridian.   There 
are  two  of  these,  as  follows :  x 

a]  The  Compass-Deviation.— Ry  this  is  meant  the  reduction  of  the  cor- 
rected pointing  to  the  Magnetic  Meridian,  whatever  the  place  or  head- 
ing of  the  ship,  and  whether  the  ship  be  upright  or  in  a  state  of  heel 
(Chap.  IV). 

b)  The  Magnetic  Variation. — This  is  to  be  understood  as  the  reduc- 
tion of  the  Magnetic  Meridian  at  the  time  and  place  to  the  True  Meridian 
(Chap.  II). 

These  are  sometimes  of  the  same  name  and  sometimes  of  different 
names,  and  it  is  their  resultant  or  combined  effect  which  constitutes  the 
Keduction-Error  in  a  given  case. 

4.  The   Total    Error ;    called   simply  the  Compass- 
Error. — Each  of  the  errors  heretofore  mentioned,  whether  belonging 
to  the  first  or  second  class,  admits  of  being  measured  on  the  compass- 
card;  and  each,  according  as  it  has  the  effect  to  carry  the  zero  or  N 
point  of  the  card  to  the  eastward  or  westward,  is  easterly  and  marked 
E,  or  westerly  and  marked  W.    Hence,  as  in  other  analogous  cases,  if 
all  the  individual  or  component  errors  have  the  same  name,  their  sum 
is  the  total  error,  and  takes  the  common  name  of  its  several  compo- 
nents ;  but  if  a  part  of  the  components  have  one  name,  and  a  part  the 
other  name,  then  the  difference  of  the  two  partial  sums  is  the  total 
error,  and  takes  the  name  of  the  greater  partial  sum. 

The  total  or  combined  effect  of  the  several  component  errors  of  a 
compass  will  be  called  the  Compass- Error,  and  will  always  be  under- 
stood to  represent  the  angular  distance  east  or  west  from  the  True  Me- 
ridian to  the  observed  compass-pointing  as  read  off  from  the  compass. 
It  may  therefore  be  defined  in  two  different  ways,  as  follows : 

a)  The  Compass-Error  is  the  angle  formed  at  the  centre  of  the  com- 
pass, between  the  N  and  S  or  zero  line  of  the  card  and  the  True  Merid- 
ian ;    and  it  is  marked  J<7  or  W,  according  as  the  N  or  zero  point  of 
the  card  falls  to  the  eastward  or  westward  of  the  true  north ;  otherwise, 

b)  The  Compass-Error  is  the  difference  between  the  compass-bearing 
and  the  true  bearing  of  the  same  object ;  and  it  is  marked  E  or  W, 
according  as  the  compass-bearing  falls  to  the  left  or  right  of  the  true 
bearing,  the  observer's  eye  being  supposed  at  the  centre  of  the  compass. 


FINDING    THE    COMPASS-ERROR. 


§  2.— FINDING  THE  COMPASS  ERROR. 

•  There  are  several  different  methods  of  Finding'  I  lie  C'oiiipass- 

JError,  by  which  will  always  be  meant  the  total  or  resultant  error,  as 
defined  in  the  preceding  articles.  It  is  one  of  the  most  important  prob 
16ms  with  which  the  Navigator  has  to  do  ;  since,  how  important  soever 
the  problems  of  the  Position,  no  accurate  steering  of  .a  given  course, 
from  one  position  to  another,  can  be  had  without  a  reliable  determina- 
tion of  the  Compass-Error  on  that  course. 

5.  The  Basis  of  Procedure  in  all  Cases.— To  find  the  Com- 
pass-Error in  a  given  case  depends  on  the  possibility  of  knowing  the 
True  Azimuth  of  some  celestial  or  terrestrial  object,  whenever  the  Com- 
pass-Azimuth of  the  same  object  can  be  had  by  observation.     Then — 

The  Compass-Error  is  found  by  taking  the  difference  between  the 
Compass- Azimuth  and  the  True  Azimuth,  and  is  marked  E  or  JF,  accord- 
ing as  the  Com  pass- Azimuth  falls  to  the  left  or  right  of  the  True  A/.i- 
muth  (4). 

The  problem  will  therefore  consist  of  two  parts :  First,  of  finding  by 
actual  observation  the  Compass- Azimuth  of  some  object;  and,  secondly, 
of  finding,  by  computation  or  otherwise,  the  True  Azimuth  of  the  same 
object,  if  not  already  known  by  a  previous  determination.  If  the  object 
be  a  celestial  body,  in  motion,  the  True  Azimuth  must  be  found  for  the 
instant  of  time  at  which  the  Compass-Observation  is  made;  but  if  the 
object  be  a  terrestrial  one,  at  rest,  it  is  obviously  not  important,  so  far 
as  the  mere  result  is  concerned,  whether  the  True  Azimuth  be  found  at 
the  time  it  is  required,  or  happens  to  be  already  known,  provided  the 
observer's  position  remains  unchanged. 

6.  Different    methods    of*   Finding    the    True    Azi- 
muth.— There  are  several  different  methods  of  finding  the  True  Azi- 
muth of  both  celestial  and  terrestrial  objects;   and,  accordingly,  there 
may  be  at  least  as  many  different  methods  of  finding   the  Compass- 
Error.    The  following  methods  will  be  considered  : 

I.  By  Observations  of*  Celestial  Objects. 

Method  of  Horizon-Azimuths. 
Method  of  Time- Azimuths. 
Method  of  Circumpolar  Azimuths. 
Method  of  Altitude- Azimuths. 
Method  of  Time-Altitude  Azimuths. 
Method  of  Transition- Azimuths. 

II.  By  Observations  of  Terrestrial  Objects. 

Method  of  Direct  Bearings. 
Method  of  Allignments. 
Method  of  Reciprocal  Bearings. 


DIFFERENT    METHODS    OF    FINDING    IT. 


Other  processes  of  getting  the  True  Azimuth,  incidental  for  the  most 
part  to  determinations  of  the  Time  and  Position,  are  known  to  Nautical 
Astronomers.  These,  if  made  available  by  corresponding  Compass- 
Observations,  would  supply  so  many  additional  means  of  finding  the 
Compass-Error;  but  the  methods  named  above  include  all,  probably, 
that  can  be  usefully  employed  in  a  direct  manner  for  this  purpose.1 

In  practice,  the  Navigator  may  be  required — 

First,  to  find  the  Compass-Error  on  a  particular  heading  of  the  ship, 
for  immediate  use,  upon  every  change  of  course;  or, 

Secondly,  to  find  the  corresponding  Compass-Errors  on  a  series  of  dif- 
ferent headings,  for  subsequent  use,  with  the  opportunity  chosen  at 
pleasure,  either  at  sea  or  in  port. 

In  the  first  case,  the  Compass- Err  or2  is  what  is  required,  whether  in 
shaping  a  course  or  in  working  up,  in  which  the  direct  relation  of  the 
compass-course  to  the  true  course  is  always  sought ;  but,  in  the  second 
case,  the  Com  pass- Errors  are  generally  required  to  be  reduced  to  Compass- 
Deviations?  by  separating  the  magnetic  variation,  and  also  the  Pointing- 
Error  if  required  (Chap.  IV). 

'The  method  of  Time- Altitude-Azimuths,  included  in  the  foregoing  list,  requiring 
both  the  Altitude  and  Hour-Angle  as  data,  belongs  rather  to  the  class  of  incidental 
methods,  as  it  could  hardly  ever  seem  to  be  expedient  to  use  this  as  a  direct  method 
in  preference  to  the  Time-Azimuth,  although  it  is  sometimes  so  employed. 

2  Still  called  the  "  Variation"  in  most  of  the  books  on  navigation. 

3The  reduction  of  Compass-Errors  to  Compass-Deviations  by  separating  the  Mag- 
netic Variation  has  been  fully  explained  in  a  previous  Chapter.  It  may,  however,  be 
best  to  repeat  so  much  of  it  as  may  suffice  to  insure  that  this  important  operation  shall 
always  be  performed  with  certainty. 

1.  Recall  an  application  of  the  rule  of  Art.  4: — That,  when  the  Var.  and  Dev.  have 
the  same  name  their  Sum  is  the  Error,  which  takes  the  name  common  to  both  ;  but 
that,  if  they  have  unlike  names,  their  Difference  is  the  Error,  which  then  takes  the  name 
of  the  greater  quantity. 

2.  Conceive  the  Error  and  Var.  to  be  laid  off  on  the  compass-card,  beginning  at 
the  N  or  zero  point,  and  extending  East  or  West,  according  as  they  are  marked  E  or  W, 
and  measured  on  the  arc  of  reading.     Then  the  Dev.  is  always  equal  to  the  Difference 
between  the  points  of  reading  of  the  Error  and  Var.  (i.  e.,  equal  to  their  Sum  or  Difference, 
according  as  they  have  unlike  or  the  same  names),  and  is  marked  E  or  W,  according  as 
the  Var.  falls  to  the  left  or  right  of  the  Error. 

Always  mentally  test  the  result  by  the  first  rule. 


FINDING    THE    COMPASS-ERROR. 


I. — BY   OBSERVATIONS   OF   CELESTIAL   OBJECTS. 
• 

A.  METHOD  OF  HoRIZON-AfclMUTHs, 

7.  Fundamental     Principles    of"    the    Horizon -Azi- 
muth.— The  Metbod  of  Horizon-Aziinutbs  consists  in  observing  tbe 
compass-bearing  of  tbe  Sun  or  otber  celestial  body,  wbile  its  centre  is 
in  tbe  True  Horizon;  that  is  to  say,  either  at  the  True  Rising  or  True 
Setting  of  the  body. 

But  since,  during  an  observation,  the  object  can  only  be  referred  to 
the  Apparent  or  Visible  Horizon,  and  since,  moreover,  it  is  subject  to 
vertical  displacement  from  refraction,  parallax,  and  dip,  it  is  necessary 
to  adopt  some  provision  by  means  of  which  the  actual  observation  in  a 
given  case  may  be  made  to  satisfy  the  preceding  condition.  There  are 
two  ways  of  doing  this:  one  by  observing  the  bearing  of  the  object 
while  it  is  elevated  at  a  certain  estimated  distance  above  the  Visible 
Horizon,  sufficient  to  correct  the  effect  of  the  vertical  displacement 
above  mentioned ;  the  other,  by  observing  the  body  when  its  centre  is 
apparently  in  the  Visible  Horizon,  and  applying  a  suitable  correction. 

The  former  of  these  two  provisions  is  that  which  is  more  commonly 
in  use,  at  least  by  American  Navigators;  but,  apart  from  the  uncer- 
tainty of  the  estimated  height,  it  wholly  excludes  the  Moon  as  one  of 
the  objects  of  observation.  The  latter,  on  the  other  hand,  not  only 
admits  of  a  more  precise  observation,  but  readily  permits  the  use  of 
the  Moon  as  an  object  certainly  only  inferior  to  the  Sun  in  importance 
for  this  purpose. 

8.  Remark  :   The  Amplitude.— The  method  now  being  con- 
sidered is  commonly  treated  in  the  books  on  Navigation  as  an  Ampli- 
tude; that  is  to  say,  by  using  the  complement  of  the  Azimuth,  reckoning 
from  the  E  and  W  points  toward  the  N  or  S.    But  the  Azimuth,  besides 
being  more  natural  and  more  convenient  in  the  compass-reading  than 
the  Amplitude,  has  the  advantage  of  being  consistent  with  all  the  other 
forms  of  finding  the  Compass-Error.1 

'The  author  may  bo  deemed  a  little  hardy  iti  venturing  to  propose  the  change  indi- 
cated in  the  text.  But,  so  far  as  mere  terms  are  concerned,  why  may  not  "  Kising- 
Azimuth"  and  " Setting-Azimuth"  replace  the  old  phraseology  of  " Rising-Amplitude" 
and  "  Setting-Amplitude"  without  doing  serious  violence  to  nautical  usage  .' 

With  the  change  proposed,  every  observation  at  sea  for  Compass-Error  will  be  an 
Azimuth,  always  referred  to  the  Meridian,  and  always  read  forward  from  the  zero-point, 
rather  than,  as  in  the  exceptional  case  of  the  Amplitude,  referred  to  the  prime  vertical 
and  read  backward  from  90°. 

The  Table  of  Horizon- Jzimulhs  (XXIV)  here  given  will  at  least  enable  the  Naviga- 
tor to  test  this  question  from  a  practical  point  of  view  ;  but  any  one,  still  desirous  of 
preserving  the  old  form,  may  treat  his  observation  as  an  Amplitude,  by  simply  using 
the  difference  between  90°  and  the  quantity  taken  from  this  Table,  which  is  the  same 
a*  that  taken  from  a  Table  of  Amplitudes. 


METHOD    OF    HORIZON-AZIMUTHS. 


9.  Rule  :  To  make  the  Observation.— In  observing  objects 
Laving  appreciable  disks,  like  the  Sun  and  Moon,  no  preparation  is 
necessary  beyond  that  of  being  ready  at  the  compass  a  few  minutes  in 
advance,  and  keeping  the  sight- vanes  pointed  in  the  proper  direction. 

a)  Observation  of  the  Sun. — At  the  rising,  when  the  upper. limb  ap- 
pears in  the  sea-horizon,  take  the  bearing  by  compass ;  and  continue  to 
take  bearings  of  the  centre,  bisecting  the  disk,  and  reading  off  each 
bearing  for  an  assistant  to  note,  till  the  lower  limb  appears.  Or,  at  the 
setting,  when  the  lower  limb  touches  the  horizon,  proceed  in  the  same 
manner  till  the  upper  limb  disappears.  In  either  case  take  the  mean  of 
the  several  bearings,  which  will  be  the  apparent  Compass-Azimuth  of 
the  Sun's  centre. 

6)  Observation  of  the  Moon. — Proceed  in  the  same  manner  as  in 
observing  the  Sun,  whenever  the  luminous  portion  of  the  disk  is  suffi- 
ciently large  and  bright  for  the  rising  to  be  anticipated  in  time  to  get 
the  bearing  of  the  upper  limb  as  it  comes  upon  the  horizon.  Other- 
wise, if  the  brightness  be  insufficient  for  a  reliable  observation  in  the 
horizon  (as  when  the  Moon  is  near  her  conjunction),  it  may  be  better 
to  make  the  observation  at  a  convenient  altitude  by  one  of  the  methods 
to  be  described. 

c)  Observation  of  a  Planet  or  Fixed  Star. — It  is  seldom  practicable  to 
obtain  a  reliable  observation  of  a  Planet  or  Fixed  Star  in  the  horizon, 
and  especially  at  rising;  so  that  it  is  generally  better  to  anticipate  the 
setting,  or  delay  until  after  the  rising,  and  take  the  object  low  in  alti- 
tude by  the  method  of  Time- Azimuths.  Whenever  the  attempt  is  made 
to  obtain  a  Bising-Azimutb,  it  will  be  expedient  to  prepare  for  the  obser- 
vation as  follows : 

With  the  Lat.  in,  and  the  Dec.  taken  from  the  Naut.  Almanac,  or 
from  Tab.  LX,  find  the  Time,  Apparent  or  Mean,  of  rising  (Tab.  XXIII 
and  Introd.}  Also  take  out,  from  Tab.  XXIV,  the  Horizon-Azimuth. 

Then,  a  few  minutes  before  the  time  of  rising,  having  applied  the 
latest  known  Compass- Error  to  the  Horizon  Azimuth  for  the  particular 
heading  of  the  ship,  and  thus  obtained  the  approximate  Compass-Azi- 
muth, place  the  sight  vanes  upon  the  indicated  bearing  and  watch  for 
the  Planet  or  Star  to  appear;  and  at  the  instant  it  does  come  distinctly 
into  view,  on  or  extremely  near  the  Visible  Horizon,  carefully  take  the 
bearing.1 

1O.  Remark:  Ofoier rations  auxiliary  to  Compass- 
Azimuths. — There  arc  certain  auxiliary  observations  which  should 
always  be  made  at  the  instant  of  taking  any  Azimuth-Observation  for 
Com  pass- Error.  These  are — 

i)  The  Ship's  Heading  by  the  Standard  or  Azimuth  Compass,  and  by 
each  of  the  other  compasses  on  board  which  are  suitably  mounted  for 
such  observations;  and 


1  In  this  case,  of  course,  but  a  single  bearing  can  be  taken,  and  greater  care  is  there- 
fore requisite. 


FINDING    THE    COMPASS-ERROR. 


2)  If  the  ship  be  iron-built,  the  Angle  of  Heel  by  clinometer,  and 
whether  to  starboard  or  port  (Chap.  IV). 

The  labor  of  making  and  recording  these  auxiliary  observations  is 
trifling;  and  they  are  essential  to  a  complete  knowledge  of  the  compass- 
situation  (Chap.  IV). 

11.  Rule:    To    correct  the  Observed    Azimuth.— As 

already  remarked  (7),  all  observed  Azimuths  in  the  Apparent  Hori/on 
require  certain  corrections,  in  order  to  convert  them  into  Oompass-Azi 
muths  on  the  True  Horizon.     For  this  correction,  enter  Tab.  XXVII, 
with  the  Lat.  and  Dec.,  both  to  the  nearest  degree,  and  when  taken  out 
apply  it  to  the  observed  Azimuth  in  the  following  manner: 

a)  For  the  Sun,  a  Planet,  or  Fixed  Star. — At  the  rising  in  N  Lat.  and 
setting  in  S  Lat.,  apply  the  whole  correction  to  the  right;  or  at  the  set- 
ting in  N  Lat.  and  rising  in  S  Lat.,  apply  it  to  the  left. 

b)  For  the  Moon. — Apply  the  7m?/-correction  in  the  contrary  manner 
The  result  in  either  case  will   be  the   proper   Compass-Azimuth, 

which,  by  comparison  with  the  True  Azimuth,  will  give  the  Compass- 
Error. 

12.  Examples  of  Correcting  Azimuths  ohserved  in 
the  Apparent  Horizon. 


Ex.  1.— With  Lat.  55°  N  and  Dec.  14° 
N,  observed  Rising-Azimuth  of  the  Sun 
was  N  47°  E. 
By  Tab.  XXVII: 

Obs.  Az.        N  47°.o  E 
Corr.  (R.)  i  .o 

COMP.  Az.    N  48  .o  E 

Ex.  2.— With  Lat.  47°  N,  Dec.  27°  S, 
observed  Setting-Azimuth  of  Vertus  was 
S  450-5  W. 


Ex.  3.— With  Lat.  65"  S  and  Dec.  n° 
S,  observed  Rising-Azimuth  of  the  Snn  was 
S  56°  E. 

By  Tab.  XXVII: 

Obs.  Az.        S  56°.o  E 
Corr.  (L.)  i  .6 

COMP.  Az.    S  57  .6  E 

Ex.  4.— With  Lat.  350.6  S,  Dec.  20^.2 
N,  observed  Setting-Azimuth  of  the  Moon 
was  N  7o°.2  W. 


Obs.  Az.        S    45G-5  W  Obs.  Az.        N    70  .2  W 


Corr.  (L.)  o  .9 

COMP.  Az.    N  135  .4  W 


Corr.  (|  L.)          o  .3 
COMP.  Az.    S  129  .5  W 


It  will  be  noticed,  in  cases  where  the  Lat.  and  Dec.  have  contrary 
names,  in  which  it  maybe  convenient  to  take  the  supplement  of  tbe 
Compass-Azimuth  for  comparison  with  the  True  Azimuth,  as  in  Exam- 
ples 2,  3,  4,  5,  and  6  (14),  that  the  result  is  the  same  whether  the  cor- 
rection bo  applied  before  or  after  taking  the  supplement. 

Of  course,  when  the  correction  is  quite  small  it  may  be  neglected,  as 
it  might  be  in  Ex.  i  of  Art.  14. 

13.  Rule  :  To  find  the  True  Azimuth.— For  this,  proceed 
as  follows : 

a)  Preparation  of  Data.— Find  the  Latitude  in,  and  take  out  the  Decli- 
nation, both  to  the  nearest  tenth  of  a  degree.1  For  the  latter,  observe 
that— 

'It  will  be  seen  that  little  use  is  made  by  the  author  of  sexagesimal  computations, 
tabular  or  otherwise,  in  anything  relating  to  the  Azimuth  and  Compass-Error.  In 


METHOD    OF   HORIZON-AZIMUTHS. 


1.  For  the  Sun,  it  is  sufficient  to  take  it  off-hand  from  Tab.  LX. 

2.  For  a  Fixed  Star,  it  may  be  taken  at  sight  from  Tab.  LXI. 

3.  For  one  of  the  four  Planets,  it  is  sufficient  to  take  it  off-hand  from 
the  Tables  of  the  Nautical  Almanac. 

4.  For  the  Moon,  it  will  be  sufficient  to  find  the  Greenwich  date  to 
the  nearest  tenth  of  an  hour,  and  with  this  take  out  the  Declination 
from  the  Nautical  Almanac. 

b)  Solution  ~by  Computation :   Use  of  Tables  X  and  XI. — To  the  secant 
of  the  Lat.  add  the  sine  of  the  Dec.;  their  sum,  rejecting  10  from 
the  index,  is  the  cosine  of  an  angle.     This  taken  at  sight  from  Tab.  X 
to  the  nearest  tenth  of  a  degree,  if  the  Lat.  and  Dec.  have  the  same 
name,  will  be  the  required  True  Azimuth  less  than  90°;  but,  if  they 
have  contrary  names,  the  supplement  of  the  angle,  or  that  found  by  sub- 
tracting the  angle  from  180°,  will  be  the  True  Azimuth  greater  than  90°. 
In  either  case,  mark  the  Azimuth  1ST  or  S,  according  to  the  Lat.,  and  E  or 
W,  according  as  it  is  found  for  a  rising  or  setting  of  the  object  observed. 

c)  Examples  of  True  Azimuths  by  Computation  : 


Ex.  1.— With  Lat.  42°  25'  N  and  Dec. 
21°. 7  N,  what  is  the  True  Azimuth  at 
Rising? 

Lat.  42° .4  N     8600.1317 

Dec.  21  .7  N     sin  9.5679 

T.  Az.     N  60  .0  E     cos  9.6996 

Ex.  2.—  With  Lat.  51°  30'  N  and  Dec. 
16°  17'  S,  what  is  the  True  Azimuth  at 
Setting  ? 

Lat.  5l0-5  N      sec  0.2058 

Dec.  1 6  .3  S       sin  9.4482 

T.  Az.    N  116  .8  W    cos  9.6540 


Ex.  3 — With  Lat.  55°  5'  S  and  Dec. 
i3°-9  S3  what  is  the  True  Azimuth  at  Set- 
ting ? 

Lat.  55°. i  S      sec  0.2425 

Dec.  13  .9  S       sin  9.3806 

T.  Az.     S65'.2\V    0039.6231 

Ex.  4.— With  Lat  16°  57'  S  and  Dec. 
29°  13'  N,  what  is"  the  True  Azimuth  at 
Rising? 

Lat.  i6°.9  S     sec  0.0194 

Dec.  29  .2  N    sin  9.6883 

T.  Az.     S  120  .7  E    cos  9.7077 


fact,  at  least  in  his  opinion,  all  refinements,  such  as  the  consideration  of  quantities 
smaller  than  minutes  of  arc  and  tenths  of  a  minute  in  time,  are  a  waste  of  time  and  pa- 
tience, being  utterly  useless  in  computations  of  this  kind.  And,  moreover,  so  far  as 
the  finding  of  any  single  Compass-Error  is  concerned,  it  is  always  quite  sufficient  to 
work  to  the  nearest  tenth  of  the  degree  in  arc,  and  to  the  nearest  minute  in  time. 

The  practice  of  changing  sexagesimal  quantities  of  a  lower  denomination  into  tenths 
of  a  higher  is  readily  done  at  sight,  without  thinking  about  it,  after  a  little  practice. 

Take  the  series  in  minutes  of  arc— 


I'  2' 

0°.02     0°.03 

or,  o°.o     o°.o 


3 

o°.o5 
o°.o 


Again,  take  the  series — 
24'        25'        26' 


2/ 

o°-45 


5' 
o°.o8 


28' 


o°.io   to  tenths  and  hundredths  of  degrees, 
o°.i     to  the  nearest  tenth. 


29         3o' 
o°47    o°48    o°.5o    to  tenths  and  hundredths, 

o°  5      o°.5      o°.5      to  the  nearest  tenth. 


And  similarly  for  all  other  cases.  To  annex  one  cipher  to  the  minutes,  divide  by  6, 
and  point  off  two  decimal  places,  is  to  convert  the  sexagesimal  into  tenths  and  hun- 
dredths of  degrees.  Or,  to  divide  simply  by  6  and  point  off  one  decimal,  is  to  have  an 
equivalent  in  tenths.  But,  as  remarked  above,  the  operation  becomes  as  involuntary 
as  the  use  of  the  multiplication-table. 


10 


FINDING    THE    COMPASS-ERROR. 


Ex.  5.— With  Lat.  67°  i/N  and  Dec. 

19°  28'  S,  what  is  the  True   Azimuth  at 

Rising  ? 

Lat.  6;°.3  N    sec  0.4135 

Dec.  19  .5  S     sin  9-5235 

T.  Az.    N  149  .9  E    cos  9.9370 


Ex.  6.— With  Lat.  71°  55' N  and  Dec. 

12°  47'  N,  what  is  the  True  Azimuth  at 

Setting  ? 

Lat.  7i°'9  N  sec  0.5077 
Dec.  12  .8  N  sin  9.3455 
T.  \z.  N  44  .5  W  cos  9.8532 


d)  Solution  by  Inspection  :    Use  of  Tab.  XXIV.— Enter  the  Table  of 
Horizon- Azimuths  with  the  Lat.  and  Dec.,  each  to  the  nearest  tabular 
argument,  and  take  out  the  corresponding  angle ;  and  this,  or  its  sup- 
pleinent,  according  as  the  Lat.  and  Dec.  have  the  same  name  or  con- 
trary names,  will  be  the  True  Azimuth.    Mark  as  above  (6). 

e)  Remark. — If  either  the  given  Lat.  or  Dec.  be  more  nearly  half-way 
between  two  adjacent  arguments  than  equal  to  either  of  them,  take  the 
Mean  of  the  two  numbers  corresponding  to  the  two  arguments.     If 
both  the  Lat.  and  Dec.  be  more  nearly  half-way  between  adjacent  argu- 
ments than  equal  to  them,  take  the  Mean  crosswise  of  two  of  the  four 
corresponding  numbers. 

The  Mean  of  two  numbers  is  most  conveniently  found  by  adding  their 
Half-Difference  to  the  less  one. 

f)  Examples  of  True  Azimuths  by  Inspection  (the  Data  from  the  Ex- 
amples [c]  ) : 


Ex.  1.— With  Lat.  42^4  N   and  Dec. 
2i°-5  N,  Tab.  XXIV  gives- 
True  Az.  N  6o°.2  E 

Ex.   2.— With  Lat.  5i°.5  N  and  Dec. 
i6°.3  S,  we  get — 

TrueAz.  N  iiG0.;  W 

Ex.  3.— With  Lat.   55°.!   S  and   Dec. 
if. 9  S- 

True  Az.  S  65°.!  W 


Ex.  4.— With  Lat.    i6°.9  S  and  Dec. 

29°.2  N— 

True  Az.  S  I2o°.5  E 

Ex.  5.— With  Lat.   67°.3  N  and  Dec. 
190.5  S- 

True  Az.  Nfi49°.7  E 


Ex.  6. 

i2°.8  N— 


-With  Lat.   7i°.9  N  and  Dec. 
True  Az.  N  44°. 4  W 


In  Ex.  i,  on  entering  Tab.  XXIV,  we  find  that  the  Lat.  falls  more 
nearly  half-way  between  42°  and  43°  than  equal  to  either,  and  we  there- 
fore take  the  Mean  of  the  corresponding  Tabular  Azimuths,  6o°.4  and 
59°.9,  by  subtracting  their  Half-Difference  (o°.2)  from  the  greater. 

In  Ex.  2,  both  the  Lat.  and  Dec.  are  more  nearly  half-way  between 
51°,  52°  and  i6°.o,  i6°.5  respectively;  and,  consequently,  we  take  a 

crosswise  Mean  of  the  four  Tabular  Azimuths  [  /*  '°'  2  '2  > ,  by  adding 

(   03  .4,   02   .5   ). 

the  Half-Difference  (o°.i)  of  63°.4,  63^2  to  the  less,  or  subtracting  it  from 
the  greater. 

By  a  similar  mental  process,  executed  at  sight,  the  results  in  the  other 
Examples  are  obtained,  as  well  as  those  in  all  Examples  of  this  kind. 

It  is  thus  seen  that  with  no  greater  care  than  to  take  the  Means  of 
the  Tabular  Azimuths,  corresponding  to  the  arguments  between  which 
the  given  data  fall,  the  results  in  these  Examples,  alike  for  the  higher 
and  lower  Latitudes,  do  not  differ  more  than  o°.2  from  those  obtained  by 
computation.  And  they  will  seldom  differ  as  much  as  that. 


METHOD    OF   HORIZON-AZIMUTHS. 


In  taking  Cross-Means  of  four  numbers  we  rnay  use  either  pair  ;  but 
it  will  generally  be  found  that  one  pair,  as  in  this  instance,  has  a  smaller 
difference  than  the  other,  and  we  use  that  in  preference. 

14.  Examples  of  Fipcliiig  the  Compass-Error  by  Hori- 
zon-Azimuths. 


Ex.  1. — 1875,  June  5  :  At  sea,  in  Lat.  1 1° 
29'  N,  Long.  30°  W,  about  6h  iom  A.  Mv 
the  observed  bearing  of  the  Sun  at  rising 
was  N  59°  E  :  Required,  the  Compass-Error. 

Greenwich  date,  June  4d  2O1' 

0's  Dec.  (Tab.  LX)  N  22°.5 


By  Tab.  XXIV: 
Latitude      n°.5  N 
©'s  Dec.     22  .5  N 
Obs.  Az.  N  59  .o  E 
Corr.  (R.)      o  .1 


True  Az.     N  67°. i  E 
Comp.  Az.    N  59  .1  E 


COMP.  ERROR 8  .o  E 

Ex.  2.— 1875,  May  30:  At  sea,  in  Lat. 
25°  3'  S,  Long.  22°  W,  about  6h  42™  A.  M., 
the  observed  bearing  of  the  Sun  at  rising 
was  N  71°. 5  E :  Required,  the  Compass- 
Error. 

Greenwich  date,  May  29d  2oh 
©'s  Dec.  N  2i°.7 


By  Tab.  XXIV: 
Latitude     25°.o  S 
©'s  Dec.    21  .7  N 
Obs.  Az.  S  108  .5  E 
Corr.  (L.)      o  .3 


True  Az.      S  114°.!  E 


Comp.  Az.    S  1 08  .8  E 


COMP.  ERROR 5  .3  W 

Ex.  3.— 1875,  November  27:  At  sea,  in 
Lat.  40°  27'  N,  Long.  20°  7'  W,  about  4h  43™ 
P.  M.,  the  observed  bearing  of  the  San 
at  setting  was  S  73°  W :  Required,  the 
Compass-Error. 

Greenwich  date,  November  27d  6h 
©'s  Dec.  S  2i°.2 

By  Tab.  XXIV: 

Latitude     4O°.5  N  >  ~ 

[TrueAz.    N  ii8°.5  W 
©'s  Dec.     21  .2  S  > 

Obs.  Az.  S  73  .o  W  ?  n 

>  Comp.  Az.  N  107  .7  W 
Corr.  (L.)      o  .7        S 

COMP.  ERROR .   10  .8  W 

i 

Ex.  4.— 1875,  December  18:  At  sea,  in 
Lat.  31°  35' 'S,  Long.  60°  13'  E,  about  7h  om 
P.  M.,  the  observed  bearing  of  the  Sun  at 


setting  was  N  83°  W:  Required,  the  Com- 
pass-Error. 

Greenwich  date,  December  i8(1  3^ 
©'s  Dec.  S  23°4 


By  Tab.  XXIV: 


Latitude 
0's  Dec. 


3i°.6  S     „ 

>  True  Az. 
23  .4  8 


S  62°.  i  W 


Obs.Az.  N  83  .4W 
Corr.  (R.)      o  .4 

COMP.ERROR 34  .9  W 


Ex.  5. — 1875,  July  17:  ^  sea>  in  La^- 
51°  13'  N,  Long.  33°  47'  W,  about  2h  7m 
A.  M.,  the  observed  bearing  of  the  Moon 
at  setting  was  S  62°  W :  Required,  the 
Compass-Error  ;  also  the  Deviation. 

Ship  heading  W  %  S  by  Standard  Corn- 


Greenwich  date,  July  i6d  i6h.4 
([  's  Dec.  28°.2  S 


By  Tab.  XXIV: 
Latitude      510.2  N 
<t's  Dec.      28  .2   S 
Obs.  Az.  S  62  .o  W 
Corr.(>^R.)  o  .6 


c 


N      ^     w 
m  N  „        w 


COMP.ERROR  .............  21  .3  W 

Tab.  LVL,  Mag.  Var.  37°.  7  W 
Comp.  Dev.  16  .4  E 

Ex.  0.  —  1871,  August  16:  In  port,  Disco 
Island,  in  Lat.  69°  14'  N,  Long.  53°  18'  W, 
about  8h  40™  P.  M.,  the  observed  bearing 
of  the  Sun  at  setting  was  N  6°.8  E  :  Re- 
quired, the  Compass-Error  ;  also  the  Devi- 
ation. 

Greenwich  date,  August  i6d  i2h 

©'s  Dec.  N  13°  40' 

Latitude  69°.2  N 
©'s  Dec.  13  .7  N 
Obs.Az  N  6.8E?  , 

Corr.  (L.)     2  .4       ) 


N     0 


COMP.  ERROR  .............    52  .2  W 

By  Tab.  LVI,  Mag.  Var.  72°.  5  W 
Hence,  also,  Comp.  Dev.  20  .3  E 


12 


FINDING    THE    COMPASS-ERROR. 


By  Tab.  XXIV : 

Latitude      48°.  i  N  >   ^  ^     N  Jo60  2  £ 
If 'a  Dec.      10  .7  S  $ 

Approx.  Coinp.  Er.  5°. 2  E 

Approx.  Coiiip.  Az.       N  101  .o  E 
•     or,  S    79  .o  E 

A  few  minutes  before  the  time  of  rising, 
the  compass  is  set  upon  the  approximate 
bearing  (S  79°.o  E),  the  appearance  of 
the  Planet  in  the  horizon  watched  for,  and, 
as  soon  as  distinctly  visible,  the  bearing 
carefully  taken.  In  this  case  it  is  S  8o°.5  E. 
Hence — 

Obs.  Az.   N  99°. 5  E  )  True  Az.     N  io6.°.2  E 
Corr.  (R.)       o  .8      >  Comp.Az.    N  100  .3  E 


Ex.1*.— 1875, February 2:  Atsea,inLat. 
48°  5'  N,  Long.  174°  15'  W,  about  10  P.  M., 
desired  to  get  an  observation  of  Jupiter 
at  rising  for  Compass-Error  on  present 
course  of  ship,  which  is  North. 

Preparation  : 

Approx.  Gr.  date,  February  2d  22h 
Ifs  Dec.  S  10°.  7 

14' s  Mer.  Pass.  i;11  $m  M.  T.  at  ship 
2_fs  Hour- Angle  5h  I2m  at  rising  (Tab. 
XXIII} 

Ship  M.  T.  i2h  53m  of  rising 

Mag.  Var.       i5°.S  E  (Tab.  LVI) 

Coiup.  Dev.    io.3W?Dev.    Table    for 

Comp.  Er.        5  .2  E  >     Stand.  Comp.  COMP.  ERROR 5  -9  E 

Remark. — Example  7  is  given  at  length  to  illustrate  the  procedure 
iu  such  cases,  which,  although  in  appearance  somewhat  complicated,  is 
really  very  simple  and  but  the  work  of  a  few  moments.  Still,  it  is  quite 
evident  that,  by  waiting  a  few  minutes  for  the  Planet  to  rise  above  the 
horizon,  a  bearing  then  taken  and  compared  with  a  Time-Azimuth  for 
that  instant  will  be  much  simpler,  and  generally  preferable,  except  when 
the  Ship-Time  is  very  uncertain.  In  such  case  the  Horizon- Azimuth,  as 
above,  might  be  useful. 

B.  METHOD  OF  TIME-AZIMUTHS. 

lo.  Fundamental  Principles  of  the  Time- Azimuth.— 

In  a  Time-Azimuth,  the  bearing  of  the  Sun  or  other  heavenly  body  is 
observed  with  the  compass  whenever  practicable,  the  Ship  or  Local 
Time  being  reliably  known  within  certain  limits. 

The  observed  bearing  is  itself  the  proper  Com  pass- Azimuth  of  the 
body. 

The  True  Azimuth  is  found  by  a  solution  of  the  Triangle  of  Position 
(Int.) 

The  Data  required  are — 

The  Hour-Angle,  deduced  from  the  Local  Time  of  the  observation 
-(Int.)-, 

The  Declination,  taken  from  the  Nautical  Almanac  or  Nautical  Tables 
(Int.}}  and 

The  Latitude  in,  as  brought  up  by  the  Beckoning  or  known  from  pre- 
vious observation. 

16.  The  Ship-Time. — The  Ship  or  Local  Time  upon  which  the 
Hour-Angle  depends,  is  the  important  element  in  this  problem.  There 
are  two  distinct  means  by  which  it  is  obtained,  whenever  required  on 
board  ship. 

The  first,  which  is  the  ordinary  one,  consists  in  the  use  of^a  reliable 
Comparing- Watch,  which  is  always  kept  regulated  to  tbe  Ship-Time, 
either  Apparent  or  Mean,  generally  to  the  former ;  being  set  to  that 
time,  or  the  error  noted  upon  it,  as  often  as  anew  determination  is  made 


METHOD    OF    TIME-AZIMUTHS.  13 


by  any  of  the  astronomical  methods  employed  for  that  purpose.  Then, 
whenever  required  at  any  intervening  moment  prior  to  the  next  deter- 
mination, the  Ship  T.  is  readily  deduced  from  the  W.  T.,  by  correcting 
the  latter  for  the  error  at  the  last  preceding  determination,  for  that  accu- 
mulated in  the  rate  during  the  subsequent  interval,  and  for  any  differ- 
ence of  Longitude  in  time  made  good  during  the  same  interval,  subtract- 
ing or  adding  the  latter,  according  as  it  is  W  or  E  of  Greenwich. 

The  second  means  is  that  of  deducing  the  Ship  T.  from  the  Chronome- 
ter on  board,  by  correcting  the  Chro.  T.  for  its  error  on  Greenwich  T.,  and 
then  for  the  place  of  the  ship  by  subtracting  or  adding  the  Longitude  in 
time,  according  as  it  is  W  or  E  of  Greenwich.  The  Ship  T.,  as  thus  found, 
will  generally  be  M.  T. ;  and  if  A.  T.  be  required,  it  will  be  necessary 
to  take  out  for  the  Greenwich  date  the  Equation  of  Time  and  apply  it  to 
the  M.  T. 

For  the  observations  of  the  Sun,  it  is  more  convenient  to  have  A.  T. ; 
but  for  observations  of  the  Moon,  Planets,  or  Fixed  Stars,  it  is  almost  a 
matter  of  indifference  whether  A.  T.  or  M.  T.  be  used  ;  consequently,  it 
is  generally  expedient  to  keep  one  Com  paring- Watch  at  least  regulated 
to  ship  A.  T. 

17.  Finding-  the  Compass-Error  for  a  Single  Heading 
of  the  Ship  :  Objects  which  may  be  employed.— In  finding 
the  Compass-Error  for  a  single  heading  of  the  ship,  every  recognizable 
celestial  object,  which  is  sufficiently  bright  to  be  seen  through  the  sight- 
vanes  of  the  compass,  may  be  employed  in  Time-Azimuths  with  more 
or  less  convenience.   During  the  day,  the  Sun  is  most  commonly  resorted 
to ;  next  to  it,  the  Moon  may  often  be  used  with  nearly  equal  facility, 
whether  by  daylight  or  during  the  night,  except  when  near  her  conjunc- 
tion ;  and,  besides  these,  the  planets  Yenus,  Mars,  Jupiter,  and  Saturn 
are  frequently  available  for  this  purpose.    The  brighter  Fixed  Stars  may 
be  observed  with  advantage  whenever  the  more  conspicuous  objects  are 
in  less  favorable  positions. 

In  general,  it  will  conduce  to  accuracy  to  select  an  object,  on  a  given 
occasion,  relatively  low  in  altitude,  not  only  that,  in  being  seen  by  direct 
vision  through  the  sight- vanes  rather  than  by  reflection,  the  Compass- 
Azimuth  is  more  reliable  (Chap.  I),  but  because  the  condition  is  more 
favorable  for  a  reliable  True  Azimuth  when  the  Data  are  considerably 
uncertain.  Still,  as  will  be  seen  in  the  sequel,  Time-Azimuths  are  really 
available  with  sufficient  accuracy  for  a  single  Compass-Error  at  sea 
under  a  wide  range  of  circumstances. 

1 8.  To  make  a  Single  Time- Azimuth  Observation.— 

Take  a  bearing  of  the  object  with  the  Standard  Compass,  or  preferably 
a  set  of  two  or  three  bearings,  as  quickly  as  possible,  bisecting  it  each 
time,  if  it  have  a  sensible  disk  (Sun  or  Moon),  and  noting  the  times  with  a 
watch,  whose  error  on  Ship  T.  is  known. 

Note  the  heading  of  the  ship  with  the  same  compass,  and  the  corre- 
sponding headings  with  the  other  compasses  ;  also,  the  Angle  of  Heel, 
if  the  ship  be  iron-built,  with  the  clinometer  (10). 


14  FINDING    THE    COMPASS-ERROR. 


The  observed  bearing,  or  the  Mean  if  several  be  taken,  is  the  re- 
quired Compass- Azimuth  ;  and  the  noted  time,  or  mean  if  several  be 
taken,  is  the  corresponding  W.  T.  of  observation. 

19.  Finding*  the  True  Azimuth.— This  method  of  finding 
the  True  Azimuth  consists  of  two  parts :  First,  the  Preparation  of  the 
Data,  and,  secondly,  the  Solution  of  the  Triangle  of  Position.  The  second 
part  may  be  accomplished  by  Logarithmic  Computation,  by  Azimuth 
Tables,  or  even  by  a  Graphical  Construction,  and  each  may  have  cer- 
tain advantages  and  disadvantages  as  compared  with  the  others ;  but 
the  first  part  must  necessarily  be  the  same,  and  can  neither  be  simpli- 
fied nor  abridged,  with  reference  to  one  or  another  of  the  processes 
employed  for  the  second  part. 

.30.  Rule:  Preparation  of  the  Data.— In  preparing  the 
Data  of  a  Time-Azimuth,  we  may  proceed  in  the  following  manner,  ac- 
cording as  the  object  observed  is  the  Sun,  or  one  of  the  other  heavenly 
bodies : 

a)  The  object  observed  being  the  Sun.— This  presents  the 
simplest  case.    Thus — 

1.  Find  the  Greemcich  date  for  the  Ship-Time  of  observation  to  the 
nearest  hour. 

2.  Take  out  the  Declination  of  the  Sun,  with  this  date,  from  Tab.  LX, 
at  sight,  to  the  nearest  tenth  of  the  degree  5  and  note  the  Pol.  Dist.  and 
Co-Lat.,  each  to  the  nearest  tenth  of  the  degree. 

3.  Find  the  Hour- Angle  of  the  Sun. — For  this,  convert  the  Watch-Time 
of  observation  into  ship  Ap.  Time,  by  applying  the  Watch-Error  on  that 
time.    Or,  if  the  Watch-Error  be  known  only  on  ship  M.  Time,  apply  this 
error,  and  in  addition  the  Eq.  of  Time,  taken  at  sight  from  Tab.  LIX,  to 
the  nearest  tenth  of  a  minute,  according  to  its  sign,  adding  if  -+-,  sub- 
tracting if — .    Then,  if  the  observation  be  P.  M.,  the  ship  A.  T.  so  found 
is  the  Hour- Angle  West  5  or,  if  the  observation  be  A.  M.,  the  ship  A.  T. 
subtracted  from  i2h  gives  the  Hour-Angle  East. 

b)  The  object  observed  being  the  Moon,  a  Planet,  or 
Fixed  Star. — For  these  objects  there  is  a  little  more  to  be  done  in 
getting  the  Hour- Angle;  still,  the  process  of  finding  this  datum  is  the 
same  for  all  objects  except  the  Sun,  and  requires  less  time  to  execute 
than  to  describe. 

1.  Find  the  Greenwich  dale,  in  the  usual  manner : — For  the  Moon,  to  the 
nearest  minute  ;  and   for  either  of  the  Planets  or  a  Fixed  Star,  to  the 
nearest  tenth  of  the  hour. 

2.  TaTceoutthe  Declination  of  the  object,  at  sight,  to  the  nearest  tenth 
of  the  degree;  for  the  Moon  and  Planets  from  the  Naut.  Almanac,  and 
for  the  Fixed  Stars  from  Tab.  LXI ;  and  note  the  Pol.  Dist.  and  Co-Lai., 
each  to  the  nearest  tenth  of  the  degree. 

3.  Find  the  Hour-Angle  of  the  object. — For  this,  convert  the  Watch-Time 
of  observation  into  ship  A.  T.by  applying  the  Watch-Error  on  that  time. 


METHOD    OF    TIME-AZIMUTHS.  15 


Xext,  to  the  ship  A.  T.,  expressed  astronomically,  add  the  Eight  Ascen- 
sion of  the  True  Sun  (taken  from  Tab.  LIX  and  reduced  to  the  Green- 
wich date  by  Tab.  LXIV),  rejecting  24^  from  the  sum  if  greater  than 
24h,  and  the  result  will  be  the  Eight  Ascension  of  the  Meridian.  Then, 
the  difference  between  this  B.  A.  and  the  E.  A.  of  the  object  (taken  from 
the  Naut.  Aim.  or  from  Tab.  LXI  and  reduced  by  Tab.  LXIV  or  LXV), 
will  be  the  Hour-Angle,  which  will  be  W  or  E  according  as  the  latter 
B.  A.  is  less  or  greater  than  the  former.  If  the  H.  A.  thus  found  be 
greater  than  i2h,  subtract  it  from  24^  and  take  the  remainder  as  the 
proper  H.  A.  with  the  contrary  name. 

Otherwise,  if  the  Watch-Error  be  known  only  on  ship  M.  T.,  convert  the 
Watch-Time  of  observation  into  ship  M.  T.  by  applying  the  Watch-Error 
on  that  time.  Next,  to  the  ship  M.  T.,  expressed  astronomically,  add  the 
Right  Ascension  of  the  Mean  Sun  (taken  from  Tab.  LIX  and  reduced  to  the 
Greenwich  date  by  Tab.  LXVI),  rejecting  24h  from  the  sum  if  greater 
than  24b,  and  the  result  will  be  the  Eight  Ascension  of  the  Meridian. 
Then  the  difference  between  this  B.  A.  and  the  E.  A.  of  the  object  will 
be  the  Hour- Angle,  as  explained  in  the  preceding  paragraph. 

4.  Remark. — It  is  always  sufficient  to  take  out  the  Bight  Ascensions 
and  to  deduce  the  corresponding  Hour- Angles  to  the  nearest  tenth  of 
the  minute ;  and  Tables  LXIV-LXVI  afford  the  requisite  facilities  for 
taking  out  the  proper  corrections,  at  sight,  to  reduce  the  Tabular  Bight 
Ascensions1  to  the  given  Greenwich  date. 

1  In  order  to  facilitate  the  taking  out  of  Right  Ascensions,  to  the  nearest  tenth  of 
the  minute,  without  the  necessity  of  consuming  time  and  patience  in  the  requisite  in- 
terpolations, the  several  Tables  referred  to  in  the  text  have  been  constructed.  The 
following  explanations  of  their  use  may  be  convenient : 

1.  To  take  out  tlie  E.  A.  of  the  True  Sun.— Enter  Tab.  LIX  with  the  day  of  G.  date  and 
take  out  the  corresponding  R.  A.  and  the  Daily  Diff.  between  that  R.  A.  and  the  next 
following ;  or,  enter  the  Naut.  Aim.  with  the  day  of  date  and  take  out  the  R.  A.,  noting 
the  adjacent  Hourly  Diff.;  then,  in  either  case^  enter  Tab.  LXIV  with  the  Diif.  and  the 
hour  (including  tenths,  if  any)  of  the  G.  date,  and   take  out  the  corresponding  correc- 
tion, which  add  to  the  R.  A.  for  the  clay,  and  the  result  will  be  the  required  E.  A.  True 
Sun  reduced  to  the  date. 

2.  To  take  out  the  R.A.  of  the  Mean  Sun.— Enter  Tab.  LIX  with  the   day  of  G.  date, 
and  take  out  the  R.  A.  of  the  True  Sun  for  that  day,  to  which  apply  the  corresponding 
Eq.  of  Time,  according  to  its  sign,  adding  if  -J-,  subtracting  if  — ;  or,  take  the  R.  A.  of  the 
Mean  Sun  directly  from  the  Naut.  Aim.  for  the  day  of  date  ;  then,  in  either  case,  enter 
Tab.  LXVI  with  the  hour  (and  tenths,  if  any)  of  the  G.  date,  and  take  out  the  corre- 
sponding correction,  which  add  to  the  before-mentioned  sum,  or  R.  A.  of  the  Mean  Sun 
for  the  day  of  G.  date,  and  the  result  will  be  the  required  JR.  A.  Mean  Sun  reduced  to 
the  date. 

3.  To  take  out  the  R.  A.  of  the  Moon. — Enter  the  Tables  of  the  Naut.  Aim.  with  the  day 
and  hour  of  the  G.  date  and  take  out  .the  corresponding  R.  A.,  and  also  note  the  adja- 
cent Diff.  for  im  ;  then  enter  Tab.  LXV  with  that  Diff.  (found  at  foot)  and  the  minutes  of 
the  G.  date  and  take  out  the   corresponding   correction,  which  add  to  the  R.  A.  above 
mentioned,  and  the  result  will  be  the  required  R.  A.  of  the  Moon  reduced  to  the  date. 

4..To  take  out  the  R.  A.  of  a  Planet.— Enter  the  Tables  of  the  Naut.  Aim.  with  the  day 
of  the  G.  date  and  take  out  the  corresponding  R.  A.,  and  note  the  adjacent  Diff '.  for  ih 
with  its  sign;  then  enter  Tab.  LXIV  with  that  Diff.  and  the /four  (and  tenths,  if  any)  of  the 
G.  date  and  take  out  the  corresponding  correction,  which  apply  to  the  before-mentioned 


16 


FINDING   THE    COMPASS-ERROR. 


21.  examples  in  Finding  the  If  our- Angle. 


Ex.  1.— 1875,  July  17: 
Required,  the  Moon's   H.   A.   iu  Long. 
33°  47'  W  at  ih  37™  20*  A.  M.  Ship  A.  T. 

d.  h.  m.  h.  m. 

Ship  A.  T.  July  16  13  37      Ship  A.  T.      13  37.3 
Long,  iu  +215      R.  A.T.01      744 

Green wich  date  16  15  53 

R.  A.  Mend.    21  21.3 

R.  A.d  18  36.4 


H.  A.« 


2    44.6  W 


Ex.  2.— 1875,  February  23  : 
Required,  the  H.  A.  of  Jupiter  in  Long. 
174°  15'  W  at  ih  5m  A.  M.  Ship  A.  T. 


Ship  A.  T.  Feb.  22  13.1 
Long,  iu  -f  ii  6 

Greenwich  date  23    0.7 


h.    m. 

Ship  A.  T.     13    5.0 
H.  A.  T.0    22  25.6 


R.  A.Merid.  ii  30.6 
R.  A.  Jup.     13  59.6 

H.  A.  Jup.      2  29.0  E 


Ex.  3.— 1875,  April  25  : 

Required,  the  Hour-Angle  of  the  Moon, 
in  Long.  155°  20'  W  at  ob  15'"  30*  A.  M. 
Ship  M.  T. 

d.  Ji.  m.  h.  m. 

Ship  M.  T.  Apr.  24  12  15.5    Ship  M.  T.     12  15.5 
Long,  in  +  10  21.3    K.  A.  M.0     2  12.1 

Greenwich  date   24  22  37 

R.  A.  Mend.  14  27.6 

R.  A.  d  17  44.5 


H.  A. 


3  16.9  E 


Ex.  4.— 1875,  October  2  : 
Required,  the  H.  A.  *>f  Venus,  in  Long. 
42°  W  at  5h  45m  25s  P.  M.  Ship  A.  T. 

d.  Ji.  h.  m. 

Ship  A.  T.  Oct.      2  5.8  Ship  A.  T.        5  45.4 

Long,  in  +2.8  R.  A.  T.©    11233.9 

Greenwich  date     2  8.6 


R.  A.  Merid.  18  19.3 
R.  A.  Venus  12  45.1 

H.  A.  Venus    s  ^4-^ 


W 


Ex.  5.^1875,  April  4: 
Required,  the  H.  A.  of  Castor,  in  Long. 
80°  39'  E  at  iih  9™  P.  M.  Ship  A.  T. 

a.    h.  n.  m. 

Ship  A.  T.  April     411.2    Ship  A.  T.      n     9 
Long,  in  —     5.4    R.  A.  T.0      o  53.6 

Greenwich  date       4    5.8    T 

R.  A.  Merid.  12    26 

R.  A.  Star       7  26  6 
H.  A.  Star      4  36  o  W 

Ex.  6. — J875,  September  9  : 

Required,  the.H.  A.  of  Fomalhaut,  iu 
Long.  33°  55'  W  at  8h  20™  ios  P.  M.  Ship 
M.  T. 

d.    h  h.  m. 

Ship  M.  T.  Sept.  9    8.3    Ship  M.  T.       8  20.2 

Long,  in  +    2.3    R.  A.  M.0     n  14.2 

Greenwich  date    9  10.6    R  A  Merid— ~ 

R.  A.  Star     22  50.7 

H.  A.  Star      3  16.3  E 


R.  A.  according  to  the  sign  of  the  Diif.,  and  the  result  will  be  the  required  E.  A.  of  the 
Planet  reduced  to  the  date. 

5.  To  take  out  the  R.  A.  of  a  Fixed  Star.—  Enter  Tab.  LXI,  or  the  Naut.  Aim.,  with  the 
year  of  G.  date  and  takeout  at  sight  the  corresponding  R.  A.  to  the  nearest  tenth  of 
the  minute. 

In  taking  out  corrections  from  Tables  LXIV-LXVI,  it  will  be  sufficient  to  enter  the 
Tables  with  the  nearest  tabular  difference,  and  to  retain  the  nearest  tenth's  figure  in 
the  decimal  part  of  the  correction. 


i  In  Ex.  i :— By  Tab.     LTX 
Tab.  LXIV 


In  Ex.  2  :— By  Tab.      LIX 


Tab.      LXV 


In  Ex.  i  : — By  N.  Almanac 
'  Tab.      LXV 


In  Ex.  4  :— By  N.  Almanac, 
Tab.  LXIV 


R.  A.  True  Sun 
Red"  for  i5».9 

R.  A.  True  Su)i 

[  R.  A.  True  Sun 
!  Eq.  of  Time 

R.  A.  Mean  Sun 
Red"  for  22h.6 

R.  A.  Mean  Sun 

R.  A.  Moon 
Red"  for  52m 

R.  A.  Moon 

R.  A.  Venus 

Red"  IbrS'-.e 

7.'.  A.  Venus 


h.  in. 

7  41.4  at  Greenwich  noon  July  16. 

+•  2.6  Daily  Difi'.  4™.o. 


7  .44.0  tor  Greenwich  date. 

h.  m. 

2    6.5  at  Greenwich  noon  April  24. 
4-  1.9  at  Greenwich  noon. 


2    8.4  at  Greenwich  noon. 

+  3-7 

2  12.  i  for  Greenwich  date. 

h.  m. 

18  34.4  at  Greenwich  i5h.o  July  16. 

+     2.0  Diff. 


18  36.4  for  Greenwich  date. 

h.  m. 

12  43.5  at  Greenwich  noon  Oct.  2. 
I-     1.6  Diff.  1  1  ".4  for  i1'. 


12  45.1  for  Greenwich  date. 


METHOD    OF    TIME-AZIMUTHS. 


17 


».  Rule :  Solution  by  Logarithmic  Computation.— 

Using  Tables  X,  XI,  XII,  and  XIII,  we  may  proceed  as  follows : 

Take  the  Half-Difference  and  Half-Sum  of  the  Pol.  Dist.  and  Co- 
Lat.,  also  Half  the  Hour-Angle ;  then,  to  the  sine  of  the  y2  D.  add  the 
cosecant  of  the  ^  S.  and  the  cotangent  of  the  y2  H.  A.;  the  Sum  of  these 
three  logarithms,  rejecting  tens  from  the  indices,  is  the  log  tangent  of  the 
Angle  X.  Similarly,  to  the  cosine  of  the  y2  D.  add  the  secant  of  the  y2  S. 
and  the  same  cotangent  of  the  y2  H.  A. ;  and  the  Sum  of  these  three  loga- 
rithms, rejecting  tens  from  the  indices,  is  the  log  tangent  of  the  Angle  Y, 

There  will  now  be  two  Cases  to  consider,  according  as  the  yz  S.  of  the 
Pol.  Dist.  and  Co-Lat.  is  less  or  greater  than  90°. 

First  Case :  yz  S.  of  Pol.  Dist.  and  Co-Lat.  less  than  90°.— Take 
the  Sum  or  Difference  of  the  Angles  X  and  Y?  according  as  the  Pol.  Dist. 
is  greater  or  less  than  the  Co-Lat.,  and  the  result  will  be  the  True 
Azimuth. 

Second  Case :  y>  S.  of  Pol.  Dist.  and  Co-Lat.  greater1  than 
90°. — Always  take  the  Difference  of  the  Angles  X  and  Y5  which  subtract 
from  1 80°,  and  the  result  will  be  the  True  Azimuth. 

In  either  Case,  mark  the  True  Azimuth  N  or  S  according  to  the  Lat., 
and  E  or  W  according  to  the  Hour- Angle. 

Remark. — It  may  sometimes  be  convenient  to  use  the  Supplement 
of  the  True  Azimuth  by  subtracting  from  180°  and  reversing  the  prefix 
N  or  S,  in  order  to  make  it  correspond  to  the  Compass-Azimuth  less  than 

o 

90 . 

.  Examples  of"  True  Azimuths  by  Computation. 


Ex.  1.— With  Lat.  20°  50'  N  and  Dec. 
2°.7  S,  what  is  the  True  Az.  of  the  Sun  at  ih 
2im  P.  M.  ship  Ap.  Time?  ©'s Hour-Angle 
ih  2im  W. 


Pol.  dist. 
Co-Lat. 
Diff. 
KDiff. 

Yz  Sum 


92°.7 
69  .2 

23  -5 
ii  .7 
80  .9 


Yz  H.  A.  oh  4om.5 
T.  Az.  N  i37°.4  W  : 


sin      9-3070  cos  9.9961 

cosec  0.0055  sec  0.8009 

cot      0.7488  cot  0.7488 

tan     0.0613  tan  1.5458 

X       49°.o  Y    88°.4 


Ex.  2.—  With  Lat.  40°  33'  N  and  Dec. 
N  2i°.8,  what  is  the  True  Az.  of  the  Sun 
at  5h  24in  43s  A.  M.  ship  A.  T.  ?  ©'s  Hour- 


Angle  6h  35 

Pol.  dist. 
Co-Lat. 
Diff. 
^  Diff. 
Yz  Sum 


.3  E. 

68°.2 


Yz  H.  A.  3h  i7™.6 
T.  Az.  N  67°.7  E  : 


sin   9.2085  cos  9.9943 

cosec  0.0678  sec  0.2856 

cot   9-9330  cot  9-9330 

tan  9.2093  tan  0.2129 

X    9^.2  Y  58°.5 


Ex.  3.— With  Lat.  12°  if  S  and  Dec. 
22°.9  S,  what  is  the  True  Az.  of  the  Sun 
at  8h  20™  5s  A.  M.  ship  A.  T.  ?  ©'s  H.  A. 
3h  39m-9  E. 


Pol.  dist. 

Co-Lat. 

Diff. 

Yz  Sum 
Yz  H.  A.  i 


67°.  i 

77  -7 
10  .6 

5  -3 
72  .4 
50™ 


sin      8  9655  cos  9.9981 

cosec  0.0208  sec  0.5195 

cot     0.2835  cot  0.2835 

tan     9.2698  tan  0.8011 

X       io°.s  Y    8i°.o 


T.  Az.  S  7o°.5  E  : 

Ex.  4.— With  Lat.  6°  n'  N  and  Dec. 
23°-3  S,  what  is  the  True  Az.  of  the  Sun 
at  5"  10™  P.  M.  ship  A.  T.  ?  0's  H.  A.  5" 
iom  W. 

Pol.  dist. 


Co-Lat. 
Diff. 

y2T)iff. 
Yz  Sum 

H.  A. 


83  .8 
29  -5 
14  .7 
98  -5 


T.  Az.  N  1 140.7  W: 


sin      9-4044  cos  9.9855 

cosec  0.0048  sec  0.8303 

cot      0.0955  cot  0.0955 

tan     9.5047  tan  0.91 13 

X       170.7  Y   83°.o 


1  lu  taking  out  the  cosecant  and  secant  in  this  case,  use  the  Supplement  of  %  S.  by 
subtracting  from  180° ;  otherwise,  and  more  conveniently,  use  the  excess  of  the  y2  S. 
over  90°,  taking  out  the  secant  for  cosecant  and  the  cosecant  for  secant. 


18 


FINDING    THE    COMPASS-ERROR. 


Ex.  5.— With  Lat.  23°  24'  N  and  Dec. 
23°4  N,  what  is  the  True  Az.  of  the  Sun 
at  6"  10™  P.  M.  ship  A.  T.  ?  ©'s  II.  A.  6" 
iom  WT. 

Pol.  dist.      66°.6 

Co-Lat.       .  66  .6 

Diff.  o  .o 

Yz  Diff.  o  .o 

Yz  Sum          66  .6 

Yz  H.  A.  31.  5m 

T.  Az.  X  67°4.  W  : 


sin      —oo 

cosec  

cot      

tan     — oo 

X         o^.o 


COS   0.0000 

sec  0.4010 
cot  9  9810 
tan  o  3820 
Y  67o.4 


Ex.  6.—  WithLat.2i°23'NandDec.2i° 
23'  S,  what  is  the  True  Az.  of  the  Sun  at 
4n  30™  P.  M.  ship  A.  T.  ?  ©'s  H.  A .  4h  30™ 
W. 

Pol.  dist.  m°.4 
Co-Lat.  68  .6 
Diff.  42  .8 

Yz  Diff.  21  .4 
Yz  Sum  90  .o 
Yz  H.  A.  2h  15- 


T.  Az.  Kn8°.6  W: 


sin  9.5621 

cosec  o.oooo 

cot  0.1751 

tan  9.7372 

X  28°.6 


COS 

sec  -}-oo 
cot 

tan  +00 
Y    o°.o 


Ex.  7.— With  Lat,  12°  if  N  and  Dec. 
ii0.;  N,  what  is  the  True  Az.  of  the  Sun 
at  5*  5™  P.  M.  ship  A.  T.  ?  ©'s  H.  A.  5" 
5mW. 

Pol.  dist.  78°.3 
Co-Lat.  77  .7 
Diff.  o  .6 

Yz  Diff.  o  .3        sin      7.7190   cos  o.oooo 

Yz  Sum        78  .o       cosec  0.0096   sec  0.6821 

Yz  H.  A.  2^  32m.5       cot      o.'io53    cot  0.1053 

tan      7.8339   tan  0.7874 

T.  Az.  N  8i°.i  W:         X        o°.4      Y    80^.7 

Ex.  8.— With  Lat.  72°  u'  N  and  Dec. 
29°  32'  N,  required  the  True  Az.  of  a  Star 
whose  Hour- Angle  is  9h  15™  W. 
Pol.  dist.      6o°.s 
Co-Lat.         17  .8 
Diff.  42  ,7 

Yz  Diff.          21  .3        sin      9.5602 
Yz  Sum         39  .1        cosec  0.2002 
cot 
tan 


Yz  H.  A.  4h  37m-5 
T.  Az.N  360.5  W: 


cos  9.9693 
sec  o.noi 
9.5757   cot  9.5757 
9.3361   tan  9.6551 

I2°.2         Y      24°-3 


Ex.  9.— With  Lat.  30°  f  N  and  Dec. 
9°  if  S,  what  is  the  True  Az.  of  the  Moon 
when  its  II.  A.  iso'1  13"'. 3  E  .' 


Pol.  dist. 
Co-Lat. 

Yz  Diff. 
Yz  Sum 
Yz  H.  A.  c 


99°-3 
59  -9 
39  -4 
19  .7 
79  .6 
6m.  61A 


T.  Az.  N  i74°.9  E  : 


sin       9.5278  COS  9-9738 

cosec  0.0072  sec  0.7435 

cot      1-5384  cot  1.5384 

tan      1.0734  tan  2.2557 

X       85°.2  Y    890.7 


Ex.  1O.—  With  Lat.  12°  if  N  and  Dec. 
11°  42'  S,  what  is  the  True  Az.  of  the  Sun 
at  61'  30'"  A.  M.  ship  A.  T.  ?  ©'s  II.  A.  5'' 
30™  E. 

Pol.  dist.     101°.  7 


Co-Lat. 
Diff. 
Yz  Diff. 
Yz  Sum 
Yz  H.  A. 


77  .7 
24  .o 
12.0 
89  .7 
45m 


sin   9-3179  cos  9  9904 

cosec  o.oooo  sec  2  2810 

cot  0.0570  cot  0.0570 

tan  9  3749  tan  2.3284 

X       i3°-3  Y 


T.  Az.  N  103°.  i  E  : 


Ex.  1 1.— With  Lat.  70°  50'  N  and  Dec. 
2o°.6  1ST,  required  the  True  Az.  of  the  Sun 
at  ob  20™  A.  M.  ship  A.  T.  ©'s  H.  A.  nh 

40™  E. 

Pol.  dist.  690.4 
Co-Lat.  19  .2 
Diff.  '  50  .2 

Yz  Diff.         25  .1        sin      9.6276   cos  9. 9569 

Yz  Sum          44  .3       cosec  0.1559    sec  °  •I453 

Yz  H.  A.  s1'  50™          cot      8.6401    cot  8.6401 

tan     8.4236   tan  8.7423 

T.  Az.  N4°.7E:  X         i°.s       Y     3°.2 

Ex.  12.— With  Lat.  i°  2'  S  and  Dec. 
32°. i  N,  required  the  True  Az.  of  Castor 
when  its  H.  A.  is  4h  35m.8  W. 

Pol.  dist.    i22?.i 

Co-Lat.         89  .o 

Diff.  33  -i 

Yz  Diff-          16  .5        sin      9.4533    cos  9.9817     . 

Yz  Sum  105  .5  cosec  0.0161  sec  o  5731 
cot  0.1632  cot  0.1632 
tan  9.6326  tan  0.7180 


Yz  H.  A.  2'i  17™. 9 
T.  Az.  S  i24°.oW: 


X  23°.2        Y 


Always, in  computing,  take  secant  for  cosecant  and  cosecant  for  secant 
of  the  excess  whenever  the  Half-Sum,  as  in  Ex.  4,  is  greater  than  9o°.o. 

In  the  use  of  Table  XXX,  this  distinction  is  not  presented  ;  the  nec- 
essary changes  being  provided  for  in  the  construction  of  the  Tables. 
This  is  one  among  the  other  advantages  of  tabular  inspection  in  avoid- 
ing the  mistakes  of  a  hasty  computation. 

24.  The  Tallies  of  Time-Azimuths.— Although,  under  the 
preceding  Eule,  the  computation  of  a  True  Azimuth  by  this  method  is 
really  quite  simple,  and  may  be  accomplished  in  a  few  moments  for  any 


METHOD    OF    TIME-AZIMUTHS.  19 


given  case,  still,  it  is  possible,  with  the  aid  of  special  Tables,  to  greatly 
facilitate  the  determination  of  this  important  element.  Such  is  the 
object  of  the  Tables,  in  three  parts  (XXX>  XXX.  A,  and  XXX.  _B),  of 
this  Manual. 

a)  Tab.  XXX,  or  the  First  Part,  provides  for  taking  out  Logs  A 
and  B  with  the  Lat.  and  Dec.,  or  with  the  corresponding  Co-Lat.  and 
Pol.  Dist.,  either  being  used  at  pleasure  as  arguments.  These  argu- 
ments comprise  all  Latitudes  from  o°  to  80°  north  or  south  of  the  Equa- 
tor, and  all  Declinations  from  o°  to  35°  of  the  same  and  contrary  name. 
The  arguments  are  given  at  intervals  of  one  degree  ;  but  the  Table  may 
be  entered  to  tbe  nearest  half-degree,  and  the  quantities  taken  at  sight, 
with  a  sacrifice  of  precision  that,  as  compared  with  a  computation  to  the 
nearest  tenth  of  the  degree,  is  seldom  so  great  as  io°.2,  or  ±12'; — 
much  within  the  requirements  of  practice  in  finding  a  single  Compass- 
Error.  The  Table  may,  however,  readily  be  entered  with  the  Lat.  and 
Dec.  to  the  nearest  tenth  of  the  degree,  by  a  simple  interpolation,  but 
this  is  rarely,  if  ever,  required  for  a  single  Compass-Error. 

1}  Tab.  XXX.  A,  or  Second  Part,  provides  for  taking  out  Log  C, 
with  the  Hour-Angle  as  argument.  This  argument  comprises  all  Hour- 
Angles  from  o11  to  i2h  west  or  east  of  the  Meridian,  and  is  given  through- 
out at  intervals  of  one-tenth  of  a  wi>ii^(every'six  seconds)  in  time. 

c)  Tab.  XXX.  B,  or  Third  Part,   provides  for  taking  out  the 
Angles  X  and  Y  among  the  arguments,  which  arejgiven  in  degrees 
and  tenths.    This  Table  is  entered  to  the  nearest  tabular  quantities  with 
the  two  sums  obtained  by  adding  Log  C  to  Log  A  and  to  Log  B. 

d)  It  was  intended  in  the  construction  of  these  Tables — 

First,  that  they  should  admit  of  furnishing  an^im mediate  or  off- 
hand determination  of  the  True  Azimuth,  with  every  probable  combi- 
nation of  Data,  without  a  moment  of  time  or  patience  expended  on  mere 
interpolation,  and  with  a  degree  of  precision  and  certainty  to  meet 
every  practical  requirement,  whether  in  finding  single  or  serial  Com- 
pass-Errors; and, 

Secondly,  that  they  should  be  as  compact  and  portable  as  possib!e, 
consistently  with  the  realization  of  the  foregoing  conditions.1 

1  Other  Tables  of  Time-Azimuths  Lave  been  given  under  different  forms.  Such  are 
the  following : 

1.  Sun's  True  Hearing  or  Azimuth  Tulles,  computed  for  intervals  of  four  minutes,  between 
the  Parallels  of   Latitude  30°    and    60°    inclusive :    By  JOHN  BURDWOOD,  Staff-Com- 
mander K.  N.,  London,  1866. 

2.  Tables  des  Azimuts  du  Soleil  corrcspondant  d  Vhcure  vraie  du  lord,  entre  les  parallelcs  55° 
sud  et  55°  nord  :  par  F.  LABROSSE,  Ancien  Officier  de  Marine,  Paris,  1868.    (The  explana- 
tions are  given  in  French  and  English.) 

3.  A  set  of  Azimuth-Tables  (pp.  58-107),  contained  in  Practical  Information   on  the 
Deviation  of  the   Compass,  for  the  use  of  Masters  and  Mates  of  Iron  Ships :  By  JOHN 
THOMAS  TOWSON,  F.R.G.S.,  etc.,  London,  1869. 

4.  Azimuth  and  Hour- Angle,  for  Latitude  and  Declination  ;  or  Tables  for  finding  Azimuth 
at  sea  ly  means  of  the  Hour-Angle,  in  all  navigable  Latitudes,  at  every  two  degrees  of  Dcclina- 


20  FINDING    THE    COMPASS-ERROR. 


Direct  and  Limiting  Values  of  Time-Azimuths : 
Use  of  Tab.  XXXI.— Tab.  XXXI  gives  the  Time-Azimuth  directly, 
which,  under  very  favorable  conditions  (60),  may  be  taken  out  quite 
readily  and  with  sufficient  accuracy  for  finding  the  Compass-Error  in 
numerous  special  instances ;  but  the  argum'ent-intervals  are  too  large  for 
convenient  use  in  the  general  case. 

This  Table  is  useful  in  showing  the  limits  of  the  Time-Azimuth  ;  in 
furnishing  a  check  against  serious  error,  through  a  hasty  or  inconsiderate 
procedure  in  getting  out  the  True  Azimuth  by  the  ordinary  methods ; 
and  it  is  frequently  useful  in  giving  an  approximate  True  Azimuth,  at 
sight,  as  required  in  estimating  probable  errors,  etc. 

26.  Rule  :  Solution  l>y  the  Azimuth-Tables.— The  Data 
having  been  prepared  (20),  proceed  as  follows : 

Enter  Tab.  XXX,  either  with  the  Lat.  and  Dec.,  or  with  the 
Co-Lat.  and  Pol.  Dist.,  to  the  nearest  half-degree,  and  take  out  at  sight 
Log  A  from  the  left  and  Log  B  from  the  right  page ;  next,  enter  Tab. 
XXX.  A,  with  the  Hour- Angle,  to  the  nearest  tenth  of  the  minute,  and 
take  out  Log  (7,  which  add  to  both  Log  A  and  Log  B,  calling  the  results 
Tang  X  and  Tang  Y;  then,  enter  Tab.  XXX.  B ,  and  take  out  (from 
the  arguments)  Ang.  X  with  the  Tang  X  and  Ang.  Y  with  the  Tang 
Y  ;  and,  finally,  complete  the  solution,  according  as  it  falls  under  the 
First  or  Second  Case  (Art.  22  and/ootf  of  Tab.  XXX.  B). 

b)  Remark. — When  either  the  Lat.  or  Dec.  (in  Tab.  XXX)  falls  more 
nearly  upon  the  half-degree  than  whole  degree  of  the  tabular  argu- 
ment, take  the  Mean  of  the  two  adjacent  tabular l  quantities ;  and  when 
both  the  Lat.  and  Dec.  fall  more  nearly  upon  the  half  than  whole  degree 
of  the  tabular  arguments,  take  the  Cross-Mean  of  the  four  adjacent 
quantities.2  The  same  rule  will  be  observed  in  using  the  Co  Lat.  and 
Pol.  Dist.  as  arguments. 

tion  between  ihe  limits  of  the  Zodiac,  etc. :  By  Major-General  R.  SHORTREDE,  F.R.A.S., 
London. 

Also,  a  Time-Azimuth  Diagram  :  By  HUGH  GODFRAY,  M.  A.,  London,  1858. 

This  graphical  method  applies  to  all  Latitudes,  to  all  Declinations  between  25°  N 
and  25°  S,  and  to  all  Hour- Angles  from  2h  to  iob,  E  or  W  of  the  Meridian. 

1  The  Mean  of  two  quantities  is  most  readily  found  by  subtracting  their  Half-Differ- 
ence from  the  greater,  or  adding  it  to  the  smaller  quantity.    Most  commonly,  the  Mean 
of  the  two  adjacent  quantities  in  this  Table  may  be  taken  at  sight. 

2  The  Mean  from  each  cross-pair  of  the  four  adjacent  quantities  in  this  Table  is  the 
same,  or  nearly  so,  and  either  may  be  used ;  but,  in  general,  it  will  be  seen  that  the 
quantities  of  one  cross-pair  in  each  case  are  very  nearly  equal,  and,  accordingly,  the 
Mean  of  this  pair  will  be  taken  in  preference. 


METHOD    OF   TIME-AZIMUTHS. 


21 


Examples  of  True    Azimuths   by  the  Azimuth- 
Tables  (the  data  from  the  Examples  of  Art.  23). 


•  Enter  with 


Ex.  1.— Lat.  20°.8  N,Dec.  2°.;  S,  and  H. 
A.  i'1  2im  W :  Required,  the  True  Az. 
P.  D.      Q20.5 
C.  L.      69  .o 
Diff.       23  .5 
Yz  Sum  80  .7 

Log  A  9.315  Log  B  0.785  Tab.  XXX 
Log  C  0.748  Log  C  0.748  XXX.  A 


Tan  X  0.063  Tan  Y  i. 533 
X     49°-2      Y      88^.3 
TRUE  Az.    N  i37°.5  W 


XXX.  B 


Ex.  2.— Lat.  40°.5  N,  Dec.  2i°.8  N,  and 
H.  A.  6h  35m-3  E  :  Required,  the  True  Az. 
P.  D.      68°.o 
C.  L.      49  .5 
Diff.       1 8  .5 
y2  Sum  58  .7 


9.274 
9-933 


0.280 
9-933 


9.207  0.213 

X      9°.  i  Y     58°.5 

TRUE  Az.    N  6j°.6  E 

Ex.  3.— Lat.  i2°.3  S,  Dec.  22°.9  S,  and 
H.  A.  3h  39m.9  E :  Required,  the  True  Az. 
P.  D.      67°.o 
C.  L.      77  .5 
Diff.       10  .5 
)^  Sum  72  .2 

0.514 

0.284 


9.266 
io°.5 


0.798 

sr.o 


TRUE  Az.    S  7o°.5  E 

Ex.  4.— Lat.  6°.2  N,  Dec.  230.3  S,  and 
H.  A.  5h  10™  W:  Required,  the  True  Az. 
P.  D.      ii3°.5 
C.  L.        84.0 
Diff.         29  .5 
Y2  Sum    98  .7 


9.410 
0.096 


0.803 
0.096 


9-506  0.899 

X       i7°.8  Y     82°.8 

TRUE  Az.    Nii5°.o\V 


Ex.  5.—  Lat.  23°.4  N,  Dec.  23°.  4  N,  and 
H.  A.  6h  iom  W  :  Required,  the  True  Az. 

P.  D.  66°-5 
C.  L.  66  .5 
Diff.  o  .o 

y2  Sum  66  .5 

—  oo  0.399 

9-98i 

—  oo  0.380 
X     o°.o                   Y     6;°.4 

TRUE  Az.    N  67°.4  W 

Ex.  6.—  Lat.  2i°4  N,  Dec.  2i°-4  S,  and 
H.  A.  4h  30™  W  :  Required,  the  True  Az. 
P.  D.    iu°.5 
C.  L.      68  .5 
Diff.       43  .o 
y2  Sum  90  .o 


9.564 
0.175 


-f-  oo 


9-739  +<* 

X      28°.  7  Y     90°.o 

TRUE  Az.    N  u8°.7\V 

Ex.  7.—  Lat.  i2°.3  N,  Dec.  ii°.7  N,  and 
H.  A.  511  5m  W  :  Required,  the  True  Az. 
P.  D.      78°.s 
C.  L.      77  .5 
Diff.         i  .o 
y2  Sum  78  .o 


7-95° 
0.105 


0.682 
0.105 


8.055  0.787 

X      o°.7  Y     80°.  7 

TRUE  Az.    N  8i°4  W 

Ex.  8.— Lat.  72°.2  N,  Dec.  29°.$  N,  and 
H.  A.  9h  15™  W  :  Required,  the  True  Az. 
P.  D.      6o°.5 
C.  L.      18  .o 
Diff.       42  .5 
*/2  Sum  39  .2 


9.758 


0.080 
9-576 


9-334  9-656 

X         I2°.2  Y        24°.3 

TRUE  Az.    N  36°-5  W 


22 


FINDING    THE    COMPASS-ERROR. 


Ex.  9. — Lat.  30°. i  N,  Dec.  9°. 3  S,  and 
H.  A.  oh  i3m-3  E :  Required,  the  True  Az. 
.P.  D.      99°.5 
C.  L.      60  .o 
Diff.       39  .5 
Yi  Sum  79  .7 


1-537 
I-Q73 


0.723 
1-537 


2.260 

X      85°.2  Y     89°.7 

TRUE  A z.  N  i74°-9E 

Ex.  1O.— Lat.  12^.3  N,  Dec.  ii°.7  S,  and 
H.  A.  5h  30™  E^:  Required,  the  True  Az. 
P.  D.    ioi°.5 
C.  L.      77  .5 
Diff.       24  .o 
Yz  Sum  89  .5 

9.318 
0.057 


2.050 
0.057 


9-375 


2.107 

X      i3°.3      .  Y     89°.6 

TRUE  Az.   N  io2°.9  E 


Ex.  1 1.— Lat.  7o°.8  N,  Doc.  2o°.6  N,  and 
II.  A.  uh  40™  E  :  Required,  the  True  Az. 
P.  D.      69°.5 
C.  L.      19  .o 
Diff.       50  .5 
YL  Sum  44  .2 


9.786 
8.640 

8.426 

X      i°.s  Y 

TRUE  Az.    N  4°.6  E 


O.IOO 

8.640 

8.740 


Ex.  12.— Lat.  i°.o  S,  Dec.  32°. i  N,  and 
H.  A.  4h  35m.8  W  :  Required,  the  True  Az. 
P.  D.      I22°.o 
C.  L.        89  .o 
Diff.        33  .o 
Yz  Sum  105  .5 

9-469  0.555 

0.163  0.163 

9.632  0.718 

X      230.2  Y     79°-2 

TRUE  Az.    S  i24°.o  W 


28.  Remark  I:  Oil  the  Two  Cases  of  the  Time- 
Azimiith. — In  obtaining  the  True  Azimutli  by  tbis  method,  tbe  first 
thing  to  be  noticed,  after  getting  tbe  Angles  X  and  Y,  is  tbe  relation 
of  tbe  Half-Sum  of  Pol.  Dist.  and  Co-Lat.  to  90°.  If  tbe  Half-Sum  be 
less  than  90°,  tbe  Example  falls  under  Case  I ;  if  greater,  under  Case 
II.  Thus,  of  tbe  several  preceding  Examples,  Nos.  i,  2,  3,  5,  7,  8,  9,  10, 
and  ii,  fall  under  Case  I;  Nos.  4  and  12  under  Case  II ;  while  No.  6, 
having  tbe  Half-Sum  equal  to  90°.  falls  ^indifferently  into  either  Case. 

And  the  next  thing  after  deciding  the  Case :  If  the  Example  fall  under 
Case  I,  it  is  sufficient  to  observe  whether  tbe  Pol.  Dist.  be  greater  or  less 
than  the  Co-Lat,  and  to  take,  correspondingly,  the  Sum  or  the  Differ- 
ence of  the  Angles  X  and  Y  as  the  True  Azimuth  ;  or,  if  the  Example  fall 
under  Case  II,  it  is  only  necessary  to  take  the  Difference  of  the  Angles 
X  and  Y  and  use  its  Supplement  (i.  e.  subtracting  the  Difference  from 
i8o°.o)  as  the  True  Azimuth. 

Tbe  simplicity  of  these  precepts  in  tbe  Rule  of  tbe  Time-Azimuth 
must  be  evident,  if  it  be  kept  in  mind  that  they  are  entirely  general  $ 
that  is  to  say,  are  applicable  to  the  findingjof  tbe  True  Azimuth  of  any 
celestial  object,  whatever  its  Declination,  injany  ^Latitude,  and  at  any 
hour  of  the  day  or  night. 

As  an  aid  to  the  memory,  whenever  necessary  to  recall  these  precepts 
(given  at  the  foot  of  Tab.  XXX.  B,  as  well  as  in  the  text),  tbe  following 
consideration  may  be  useful:  Observing  that,  while  "  Pol.  Dist."  is  the 
distance  of  the  object  from  tbe  elevated  pole,  Co-L:vt.  is  also'tbe  Polar  Dis- 
tance of  the  observer,  tbe  significance  of  the  fundamental  distinction  into 
Two  Cases  is  found  in  this  :  that,  accord ing]as  the  Half-Sum  orJVteau  of 


METHOD    OF    TIME-AZIMUTHS. 


the  Polar  Distances  of  the  observer  and  object  is  less  or  greater  than  90° 
(that  is,  less  or  greater  than  the  Polar  Distance  of  the  Equator),  the  prob- 
lem falls  under  Case  I  or  Case  II.  The  subordinate  distinction  into  two 
varieties  of  Case  I  is  obviously  suggestive $  while  the  Difference  and  Sup- 
plement, or  Double  Difference,  is  hardly  less  so,  when  reaching  beyond 
the  Equator  of  Case  II. 

39.  Remark  II :  On  the  Conditions  introducing  the 
Use  of  the  Symbols  — co  and  +00. — There  are  two  conditions 
of  the  Lat.  and  Dec.,  which  bring  into  view  certain  relations  of  the 
symbols — co  and  +  co ,  as  found  in  the  Azimuth-Tables,  with  respect 
to  which  a  brief  explanation  may  be  of  service  to  some  one,  possibly, 
who  may  use  these  Tables. 

a)  The  Dec.  equal  to  the  Lat.— This  condition  is  one  very  liable 
to  occur,  alike  with  the  Dec.  of  the  same  and  the  contrary  name. 

First,  when  the  Dec.  has  the  same  name  as  the  Lat. :  Then,  whether 
both  these  Data  be  entered  in  Tab.  XXX  to  the  w^hole  or  to  the  half 
degree,1  the  Log  A  is  always  —  co ;  and  as  the  Log  C  cannot  affect  it,2 
the  Ang.  X  for  —  co  is  always  o°.o, — as  in  Ex.  5. 

Secondly,  when  the  Dec.  has  the  contrary  name  :  Then,  whether  both 
these  Data  be  entered  in  Tab.  XXX  to  the  whole  or  half  degree,3  the 
Log  JB  is  always  +00  ;  and  as  the  Log  C  cannot  affect  it,4  the  Ang.  Y 
for  +  oo  is  always  90°, — as  in  Ex.  6. 

&)  The  Dec.  nearly  equal  to  the  tat. — This  is  a  condition, 
both  when  the  Dec.  has  the  same  and  when  the  contrary  name,  which  is 
of  course  more  frequently  liable  to  occur  than  the  first. 

First,  when  the  Dec.  has  the  same  name  as  the  Lat. :  Then,  the  smaller 
limit  in  Tab.  XXX  of  the  Log  A  is  —  oo  ,  and  the  adjacent  or  larger 
limit  (varying  from  7,941  to  8.025  JH  different  parts  of  the  Table) 
ahvays  differs  about  0.300  from  each  of  its  own  outward  adjacents. 
Accordingly,  if  both  the  Lat.  and  Dec.  be  entered  to  the  half-degree,  the 
Cross-Mean  is  still  to  be  taken,  as  in  Ex.  7  ;  but,  if  one  be  entered  to 
the  whole  and  the  other  to  the  half  degree,  the  proportion  is  made  to- 
wards the  —  oo  ,  by  always  subtracting  0.150  from  the  adjacent  and  larger 
limit  f  as,  for  example,  if  the  Lat.  were  i2°.o  and  Dec.  n°.5,  we  should 
take  7.800  as  the  Log  A. 

Secondly,  when  the  Dec.  has  the  contrary  name :  Then,  the  larger 
limit  of  the  Log  B  is  +00  ,  and  the  adjacent  or  smaller  limit  (varying 
from  2.059  to  1.975  in  different  parts  of  the  Table)  always  differs  about 
0.300  from  each  of  its  outward  adjacents.  Accordingly,  if  loth  the  Lat. 
and  Dec.  be  entered  to  the  half-degree,  the  Cross-Mean  is  to  be  taken, 
as  in  Ex.  10 ;  but  if  one  be  entered  to  the  whole  and  the  other  to  the 

1  Since  the  l/2  Diff.'of  the  Pol.  Dist.  and  Co-Lat.  being  o°.o,  the  Log  sine  is  — .  oo  . 

2  Except  in  the  rare  instance  of  having  the  Hour-Angle  equal  to  oh.o,  when  the  Ang. 
X  is  indeterminate. 

3  Since  the  ^  Sum  of  the  Pol.  Dist.  and  Co-Lat.  being  90°,  the  Log  secant  is  +  oo . 

4  In  any  instance  that  can  occur  in  practice. 

6  That  is,  by  subtracting  the  Half  Adjacent  Diff.  from  the  greater  limit,  which,  although 
only  approximate,  always  gives  a  result  within  about  ±0^.3. 


24 


FINDING   THE    COMPASS-ERROR. 


half  degree,  the  proportion  is  made  towards  the  +00  ,  by  always  adding 
0.150  to  the  adjacent  and  smaller  limit;1  as,  for  example,  if  the  Lat. 
were  i2°.o  and  Dec.  — n°.5,  we  should  take  2.200  as  the  Log  B. 

Although  the  preceding  explanation  of  these  relations  may  seem  at 
first  view  to  imply  it,  there  need  be  really  no  occasion  whatever  for  em- 
barrassment in  the  treatment  of  these  symbols  whenever  encountered; 
only  remembering  that  —  °o  represents  an  extremely  small  quantity  or 
nothing,  and  that  -f  oo  represents  an  extremely  large  quantity  or  infinity. 

3O.  Examples  of  Finding  the  Compass-Error  by  a 
Time-Azimuth.— The  Examples  already  given  (Arts.  21,  23,  27) 
fully  illustrate  the  process  of  getting  a  True  Azimuth  by  this  method, 
under  a  considerable  variety  of  circumstances.  It  will  be  sufficient  to 
present  a  few  more,  merely  to  exhibit  the  entire  form  of  procedure  in 
finding  the  Compass-Error  by. a  Time-Azimuth. 


Ex.  1. — 1871,  August   16:  At  auclior, 
Disco  Island,  in  Lat.  69°  14'  N,  Long.  53° 
18'  W,  at  4h  2im  46s  P.  M.,  Ship  A.  T.,  ob- 
served bearing  of  O*s  centre  was  N  56°  W  : 
Required,  the  Compass-Error. 
Greenwich  date,  August  i6d  8h 
©'s  Dec.  1 3°.  7  N :  ©'s  H.  A.  4*  2im.8  W 
Pol.  Dist.  76°»3 
Co-Lat.     20 .8 

0.128 


9.792 
0.192 

9.984 


0.192 


0.320 

X       44°-o  Y     64°.4 

True  Az.     N  io8°.4  W 
Comp.  Az.   N    56  .o  W 
COMP.  EK.        52  .4  W 
Ex.  2.— 1875,  February  23  :  At  sea,  in 
Lat.  60°  53'  N,  Long.  174°  15'  W,  at  ih  5m 
A.  M.,  Ship  A.  T.,  observed    bearing    of 
Jupiter  was  S  49°  E  :  Required,  the  Coin- 
pass-Error  and  Deviation. 
Greenwich  date,  February  23d  oh.7 
ZTsDec.  io°.7S 

It's  H.  A.  2h  29m.o  E  (Ex.  2,  Art.  21) 
Pol.  Dist.  100°.  7 
Co-Lat.       29  .1 


9.811 
0.472 

0.283 

X  62°.4 
True  A/. 
Coinp.  Az. 
COMP.  ER. 
Mag.  Var. 
COMP.  DEV. 


0.280 
0.472 

0.752 

Y     8o°.o. 
N  I42°4  E 
N  131  .o  E 
ii  .4  E 

19  .3  E  (Tdb.LVI) 
7.9W 


Ex.  3.— 1875,  April  25  :  At  sea,  in  Lat. 
50°  37'  N,  Long  155°  20'  W,  at  oh  i5m  3o3 
A.  M.,  Ship  M.  T.,  the  observed  bearing  of 
d's  centre  was  S  66°.5  E:  Required,  the 
Compass-Error. 

Greenwich  date,  April  24d  22b  37m 

d  's  Dec.  S  28°.3 

d  's  H.  A.  3h  i6»\9  E  (Ex.  3,  Art.  21) 
Pol.  Dist.  ii8°.5 
Co-Lat.       39  .5 


9.812 
0-339 
0.151 

X    54°-7 
True  Az. 
Comp.  Az. 
COMP.  ER. 


0607 
Q-339 
0.946 
Y    830.5 
N  I38°.2  E 
Nii3-5  E 
24.7  E 


Ex.  4.— 1875,  July  17  :  At  sea,  in  Lat. 
52°  38'  N,  Long.  33°  47'  W,  at  ih  37™  2O8  A. 
M.,  Ship  A.  T.,  observed  bearing  of  <[  's 
centre  was  S  66°.o  W :  Required,  the  Com- 
pass-Error. 

Greenwich  date,  July  i6d  I3h  37'" 

<[  's  Dec.  S  28°-3 

<['s  H.  A.  21'  44m.9  W  (Ex.  I,  Art.  21) 
Pol.  Dist.  ii8c.5 
Co-Lat.        52  .5 

9.738  1.029 

0.425  0.425 


0.163 
55°-5 


M54 
Y     B8°.o 
True  Az.     N  I43°.5  W 
Comp.  Az.  N  114  .o  W 
COMP.  ER.         29  .5  W 


irlhat  is,  by  adding  the  Half  Adjacent  Diff.  to  the  smaller  limit,  which,  although  only 
approximately  correct,  always  gives  a  result  within  about  ±o°.3. 


METHOD    OF    TIME-AZIMUTHS.  25 


31.  Finding  €0111  pass-Errors  for  a  Scries  of  Differ- 
ent Headings  of"  the  Ship.  —  It  is  occasionally  desirable  (Chap. 
IV)  to  determine  the  Compass-Errors  for  a  series  of  different  Headings 
of  the  Ship.  For  this,  the  Method  of  Time-  Azimuths  is  peculiarly  con- 
venient and  reliable,  whether  at  sea  or  near  the  laud. 

For  these  observations,  the  Sun  is  the  object  commonly  employed. 
Other  objects,  such  as  the  Moon,  the  brighter  Planets,  and  the  brighter 
Stars,  might  be  used,  sometimes,  with  advantage.  Generally,  the 
opportunity  may  be  chosen  at  will  ;  and  the  considerations  controlling 
the  choice  of  an  object  should  have  reference  to  the  slowness  of  its 
change  of  Declination,  the  ease  and  certainty  with  which  it  may  be 
sighted,  and  the  convenience  with  which  the  auxiliary  observations  may 
be  made.  In  all  cases,  the  object  should  be  observed  only  while  it  is  low 
in  altitude  —  never,  at  least  if  it  can  be  avoided,  when  the  H.  A.  is  less 
than  4h  ;  for  the  precision  with  which  serial  Compass-Errors  should  be 
had,  in  order  to  serve  any  really  useful  purpose,  requires  that  those  con- 
ditions be  satisfied. 


To  make  the  Observations  of  Serial  Time-Azi- 
muths. —  With  a  line  day  and  a  smooth  sea,  and  with  the  means  of 
placing  the  ship  upon  her  different  Headings  by  steaming  or  sailing, 
and  whenever  necessary,  near  the  land,  by  towing  or  warping,  the 
requisite  observations  may  be  conducted  as  follows: 

The  ship  being  steady,  take  a  bearing  of  the  Sun's  centre  with  the 
Standard  Compass,  noting  the  time  with  a  watch,  whose  error  on  Ship 
T.,  Apparent  or  Mean,  is  known  within  allowable  limits  (G).  Note  the 
Heading  of  the  ship  with  the  same  compass;  also  the  corresponding 
Headings  with  the  Steering-Compasses,  and  the  Angle  of  Heel  with  the 
clinometer  (10). 

After  changing  the  Heading  to  any  desired  extent,  and  again  steady- 
ing, repeat  the'  observation  in  all  particulars;  and  so  proceed,  changing 
the  Heading  and  repeating  the  observation,  till  the  desired  number  shall 
have  been  made,  completely  or  partially,  round  the  Compass  Circle. 

Remark  Is  Position  of  the  Object,  —  The  period  should 
be  so  chosen  that  the  Hour-  Angles  of  the  object  shall  not,  in  general, 
be  less  than  4h,  E  or  W  of  the  Meridian.  In  no  case  should  an  H.  A. 
be  less  than  3h,  E  or  W  of  the  Meridian,  except  in  cases  of  urgent 
necessity,  as  when  threatened  with  thick  weather. 

Remark  II  :  Steadying  the  Ship.—  While  the  observations 
should  be  made  with  the  least  practicable  delay,  no  flying  bearings 
should  be  taken  ;  but  the  ship  should  be  steadied  upon  the  recognized 
Heading,  not  only  during  the  time  of  making  each  observation,  but  for 
a  minute  at  the  least  before  commencing  it.1 


j  See  Cliap.  IF.— There  must  be  time  not  only  for  the  changing  induction  to  be  accom- 
plished, but  also  for  the  vibrations  of  the  compass-card  to  cease,  should  there  be  any, 
if  we  would  aim  at  getting  really  satisfactory  results. 


26  FINDING   THE   COMPASS-ERROR. 


Remark  III:  Heading-Intervals.— It  is  always  preferable 
to  make  the  observations  with  the  Headings  upon  the  regular  points  by 
compass,  thirty-two,  sixteen,  or  eight,  whenever  practicable;  but, 
when  the  circumstances  make  this  difficult  or  time-consuming,  the  results 
will  be  equally  available  if  obtained  on  any  Headings  distributed  with 
approximate  equality  round  the  whole  circle,  or  similarly  through  any 
part  of  it,  when  those  on  a  certain  part  only  are  desired  or  can  be  had. 

Remark  IT:  Care  in  Sighting  and  Reading  oil".— 
Strict  attention  should  be  given  to  Sighting  and  Beading  off.  It  should 
be  seen  that  the  Standard  Compass  is  in  nice  equilibrium.  With  a  well- 
made  and  properly-adjusted  Compass,  and  otherwise  favorable  circum- 
stances, it  should  be  practicable  to  sight  and  read  off  to  the  nearest 
quarter  of  a  degree. 

33.  Rule:  To  get  the  Serial  True  Azimuths  by  Com- 
putation.— First,  prepare  the  Data,  as  follows : 

a)  Find  the  Greenwich  date  for  the  Middle  Ship  T.  of  the  set  of  obser- 
vations, either  by  applying  the  Longitude  or  deducing  it  from  the  chro- 
nometer. 

ft)  With  this  date  take  out  the  Sun's  Declination ;  also,  the  Equation 
of  Time,  whenever  necessary  to  convert  Ship  M.  T.  into  Ship  A.  T. 

c)  Deduce  the  Co-Lat.  and  Mid.  Pol.  Dist.  each  to  the  nearest  minute 
for  observations  in  port  or  near  the  land,  or  when  the  greatest  precision 
is  desired ;  otherwise,  to  the  nearest  tenth  of  the  degree,  which  is  gen- 
erally sufficient  for  observations  at  sea. 

d)  Prepare  the  Sun's  Hour- Angles.    For  this,  convert  the  W.  Times 
of  observation  into  Ap.  Times;  then,  if  the  observations  be  P.  M.,  the 
Ap.  Times  are  the  Sun's  Hour- Angles  W;  if  the  observations  be  A.  M., 
the  Ap.  Times  subtracted  from  i2h  are  the  Sun's  Hour- Angles  E. 

Next,  proceed  with  the  computations,  as  follows : 

e)  Take  the  Half-Difference  and  Half-Sum  of  the  Mid.  Pol.  Dist.  and 
Co-Lat.;  then,  using  Tab.  IX, — 

To  the  sine  of  the  }4  D.  add  the  cosecant  of  the  y£  S. ;  the  result,  reject- 
ing 10  from  the  index,  will  be  the  Log  A.  Similarly,  to  the  cosine  of 
the  y?,  D.  add  the  secant  of  the  ^  S.  5  the  result,  rejecting  10  from  the 
index,  will  be  the  Log  B. 

/)  Now,  proceed  in  tabular  form,  by  ruling  several  vertical  columns, 
and,  dividing  each  Hour-Angle  by  2,  place  the  series  of  y2  Hour-Angles, 
in  their  order,  in  Col.  I.  Take  the  cotangent  of  each  y2  H.  A.  and  place 
it  opposite  in  Col.  II.  Add  Log  A  to  each  cotangent  and  place  the  result- 
ing Tangents  X  in  Col.  Ill ;  also,  do  the  same  with  Log  B,  and  place  the 
resulting  Tangents  Y  in  Col.  IV,  rejecting  10  from  the  index  in  each  case. 

Then,  take  out  for  each  tangent  in  Col.  Ill  the  corresponding  Angles 
X,  and  place  them  in  Col.  Y ;  also,  from  the  tangents  in  Col.  IV  the  cor- 
responding Angles  Y,  which  place  in  Col.  VI. 


METHOD    OF    TIME-AZIMUTHS.  27 


There  will  now  be  two  Cases  to  consider,  according  as  the  y2  S.  of  the 
P.  D.  and  C.  L.  is  less  or  greater  than  90°. 

g)  First  Case:  H.S.of  P.©.andC.li.Ies$than!&O°.— Take 
the  Sum  or  Difference  of  the  corresponding  Angles  X  and  Y,  according  as 
the  IP.  D.  is  greater  or  less  than  the  C.  L.,  and  place  the  resulting  True. 
Azimuths  in  Col.  VII ;  or,  in  the — 

h)  Second  Case :  H.  S.  of  P.  D.  and  C.  JL.  greater  than 

S>0°. — Always  take  the  Difference  of  the  corresponding  Angles  X  and  Y, 
which  subtract  from  180°,  and  place  the  resulting  True  Azimuths  in  CoL 
VII. 

1}  In  either  Case,  mark  the  True  Azimuth  N  or  S  according  to  the 
Latitude,  and  E  or  W  according  as  the  observations  are  East  or  West  of 
the  Meridian. 

Remark* — It  may  sometimes  be  convenient  to  use  the  Supple- 
ments of  the  True  Azimuths  (22,  e). 

34.  To  get  the  Serial  True  Azimuths  t>y  the  Azimuth- 
Tables. — The  preceding  Kule  for  computing  the  True  Azimuths  of  a 
Series,  for  precision  and  facility,  leaves  little  to  be  desired  in  an  opera- 
tion that  is  only  occasionally  encountered  by  the  Navigator  ;  neverthe- 
less, they  may  be  found  with  the  requisite  precision  and  greater  facility 
by  the  Azimuth -Tables  (XXX,  XXX.  A,  XXX.  B). 

The  Data  having  been  prepared  (33  o-d),  proceed  as  follows : 

a)  Enter  Tab.  XXX  with  the  Middle  Dec.  and  Lat.,  or  with  the  Mid. 
Pol.  Dist.  and  Co-Lat.,  both  to  the  nearest  tenth  of  the  degree,  and 
take  out  Log  A  and  Log  B,  proportioning  for  the  tenths  of  the  degree ; 
otherwise,  compute  Log  A  and  Log  B  (33  e),  as  may  be  deemed  most 
convenient. 

6)  Now,  proceed  in  tabular  form,  by  ruling  several  vertical  columns 
and  placing  the  series  of  Hour- Angles,  in  their  order,  in  Col.  I.  Then, 
entering  Tab.  XXX.  A,  with  each  H.  A.,  in  succession,  take  out  at  sight 
the  corresponding  Log  C,  and  place  it  in  Col.  II,  opposite  to  the  H.  A. 
to  which  it  belongs.  Next,  add  Log  A  to  each  Log  C  and  place  the  Sum1 
opposite  in  Col.  Ill;  also,  do  the  same  with  Log  B,  and  place  the  cor- 
responding Sums  in  Col.  IV.  Finally,  enter  Tab.  XXX.  B  with  each 
Sum  in  Col.  Ill  and  take  out  the  corresponding  Aug.  X,  which  place  in 
Col.  V;  also,  enter  the  same  Table  with  each  Sum  in  Col.  IV  and  take 
out  the  corresponding  Arig.  Y,  which  place  in  Col.  VI. 

Finally,  there  will  be  the  two  Cases  to  consider,  according  as  the  Half- 
Sum  of  Pol.  Dist.  and  Co-Lat.  is  less  or  greater  than  90°. 

c)  First  Case :  H.  S.  of  Pol.  Dist.  and  Co-Lat.  less  than  90°.— Take 
the  Sum  or  Difference  of  the  corresponding  Angles  X  and  Y,  according  as 

1  The  Sums  in  Col.  Ill  being  Tangents  X  and  those  in  Col.  IV  being  Tangents  Y. 


28 


FINDING    THE    COMPASS-ERROR. 


the  Pol.  Dist.  is  greater  or  less  than  the  Co-Lat,  and  place  the  resulting 
True  Azimuths  in  Col.  VII. 

d]  Second  Case :  H.  S.  of  Pol.  Dist.  and  Co-Lat.  greater  than  90°. — 
Always  take  the  Difference  of  the  corresponding  Angles  X  and  Y,  which 
subtract  from  180°,  and  place  the  resulting  True  Azimuths  in  Col.  ATII. 

e)  In  either  Case,  mark  the  True  Azimuths  N  or  S  according  to 
the  Lat.,  and  E  or  W  according  as  the  observations  are  East  or  West  of 
the  Meridian. 

Remark* — If  it  be  desired  to  take  the  Mid.  Dec.  and  Lat.  to  the 
nearest  minute,  it  will  be  necessary  to  proceed  according  to  Par.  e  of  the 
preceding  Rule  (33),  instead  of  Par.  a  of  the  present  Kule,  to  find  Log  A 
and  Log  B;  after  which,  proceed  according  to  the  present  Rule,  observ- 
ing that  it  will  generally  be  sufficient  to  use  Log  A  and  Log  B  with  the 
nearest  third  decimal  figure,  rejecting  the  fourth  figure. 

35.  Examples  or  Finding  Serial  Compass^Errorg  by 
the  Iflethod  of  Time-Azimuths. 

Ex.  1.— 1875,  May  13:  ID  New  York  Bay,  Lat.  40°  41'  N,  Long.  74°  2'  W,  about  5h 
45m  P.  M.,  made  the  following  observations  of  the  Suu,  on  a  careful  swinging  of  the 
Ship,  for  Compass-Deviations. 

Watch-Error  on  Loc.  M.  T.  47s  fast;  mean  from  comparisons  before  and  after  the 
observations. 

Mag.  Variation  -j°.g  W,  from  previous  determinations. 


a)  Tabular  Record  of  Observations. 


Watcb  Time. 

Standard  Compass. 

Watch-  Time. 

Standard  Compass. 

Ship's  Head. 

Q's  Centre. 

Ship's  Head. 

©'s  Centre. 

h      m        s 

5    18    10 

20      30 
22        0 

25     10 

North 
NNE 
NE 
ENE 

N  72.0  W 

64-5 
57-0 
5i-5 

h      m        s 
5    37    5° 
41      o 
46    30 
48    30 

South 

ssw 
sw 
wsw 

N  70°.  5  W 

79-5 
80.0 

89-5 

27      50 
30      20 
32      30 

34    40 

East 
ESE 

SE 

SSE 

N  49.0  W 

4&S 

p.5 
60.5 

55      o 
57    20 
6      o    30 
3     10 

West 
WNW 

NW 
NNW 

88.5 
85.5 
80.0 
73-o 

b)  Preparation  of  the  Data. 


Mid.  date  A.  T.  May    13*  5h.7  Pol.  Dist.  71°  32' 

Long.  W  -f-   4  .9  Co-Lat.  49  19 

Greenwich  date  13  10  .6  Diflf.  22  13 

YZ  Diff.  1 1  6^ 

©'s  Dec.  N  18°  2i'.3|  -f  o'.6  ^  Sum  60  25 >£ 

Redn  for  ioh.6        -f     6  .4 
0's  Dec.  for  date    18    28  N 


By  Tab.  IX 

9.2848 
0.0606 


sn 
cosec 


cos 
sec 


9.9918 
_o-3o67 


Log  A  9.3454     Log  B  0.2985 


Eq.  of  T.  3m  51*-!-  to  M.  T. 


Error  ol  Watch  on  Loc.  A.  T.  -f  3 


METHOD    OF    TIME-AZIMUTHS. 


29 


c) 

Tabular  Form  of  Solution  by  Azimuth-  Tables. 

Sun's 
Hour-An- 
gles West. 

Table 
XIX.  A. 

Log  A 
9.34514. 

LpgB 

Table  XXX.  B. 

Sun's 
True 
Azimuth. 

Swn's 
Compass- 
Azimuth. 

Compass- 
Error. 

Compasb- 
i)ev. 

TanX. 

Tan  Y. 

Ang.  X. 

Ang.  T. 

h 

m 

o 

o 

o 

o 

o 

5 

21.2 

0.074 

9.419 

0.372 

14.7 

67.02 

N8i.7W 

N;2.oW 

9-7  W 

i.8W 

23.6 

069 

414 

367 

14. 

55 

66.77 

81.3 

64-5 

16.8 

8-9 

25.1 

066 

411 

364 

14-45 

66.62 

81.1 

57-0 

24.1 

16.2 

28.2 

060 

405 

358 

14-3 

66.32 

80.6 

5^-5 

29.1 

21.2 

5 

3°-9 

0-055 

400 

353 

14 

i 

66.12 

80.2 

49-o 

31.2 

23-3 

33-4 

051 

396 

349 

14.0 

65.92 

79-9 

48.5 

23-5 

35-6 

046 

391 

344 

13-85 

65.67 

79-5 

52-5 

27.0 

I9.I 

37-7 

042 

387 

340 

13 

7 

6547 

79-2 

60.5 

18.7 

10.8 

5 

40.9 

0.036 

381 

334 

13-55 

65.17 

78-7 

70-5 

8.2  W 

0.3  W 

44.1 

030 

375 

328 

13-35 

64.87 

78.2 

79-5 

1.3  E 

9.2  E 

49-6 

020 

365 

13-05 

64.32 

77-4 

86.0 

8.6 

16.5 

51.6 

016 

361 

3H 

12.95 

64.12 

77.1 

89-5 

12.4 

20.3 

5 

58.1 

0.004 

349 

302 

12.6 

63.52 

76.1 

88.5 

12.4 

2-0.3 

6 

0.4 

9-999 

344 

297 

12.45 

63.22 

75-7 

85.5 

9.8 

17.7 

3-6 

993 

338 

291 

12 

3 

62.92 

75-2 

80.0 

4.8  E 

12.7 

6.2 

388 

333 

286 

12 

17 

62.67 

74-8 

73-o 

1.8  W 

6.1  E 

In 

taking 

out 

the  Angles  X  and  Y  it  is 

well,  in 

a  case 

like  this,  where 

considerable  precision  is  desired 

and  can  be  had,  to  allow  for  the  in- 

fluence  of  the  fourth  figure  (cut 

off)  in  Log  A  and  Log  B,  and  write  the 

nearest  second 

decimal,  as 

here  done,  in  the  values  of  those 

Angles, 

retaining 

only  the  nearest  tenth 

in 

the  values  of 

the  Azimuths. 

Ex 

.  2.  —  1866,  May  5  :  At  sea,  in  Lat.  50°  10'  N,  Long.  13° 

10'  W,  about  6h  P. 

M.,  made 

the  following  observations  of  the  Sun  for  Compass  Errors  and  Deviations,  watch  being 

fast  26m.o  on  Ship  A.  T.  : 

Observations.1 

By  Standard  Compass. 

Watch- 

Tim 

e. 

Ship's  Head. 

©'s  Centre. 

Ship  Mid. 

A.  T.  May  5d  6h_5 

Long.W 

+    I    .0 

h    m 
6  15 

N 

116  E 

N  64 

W 

Greenwich  date 

5    7  -5 

22 

40 

72 

28 

66 

75 

3 

3 

S 

88  E 
6c 

72 

hT 

0's  Dec. 

N  160 

16'  1  +o'. 

7 

°5 

°7 

Red*  for  7h. 

5        -f 

C           4  . 

a 

6  44 

S 

40  E 
6 

N  60 

W 

Red.  Dec. 

16 

J  „           «    y 
21                .3 

56 

S 

20   W 

43 

7    o 

10 

7  IS- 

N 

75 
84  W 

N23 

W 

P.  D.  730.6  c  Log  A  9.541  { 
C.  L.  39  .8*  LogB  0.242  5 

XXX 

21 

62 

21 

, 

JC 

42 

23 

*-  j 
31 

25 

o 

28 

Mag 

Var.  26° 

W 

38 

4 

36 

1  The  Data  of  this  Example  are  taken  from  Traite  de  Deviation  et  de 

Regulation  des  Com- 

pas  : 

par  E.  GIQUKL,  Professenr  d 

'hydrographie,  Paris,  1868. 

The  observations  are  said  to  have  been 

made  on  board  the  Ville  de  Paris  f  one  of  the 

ships 

of  the  New  York  and  Havre 

Line. 

30 


FINDING    THE    COMPASS-ERROR. 


Reductions  and  Results. 

Sun's 
Ho.  Ang. 
West. 

Tab. 
XXX.A. 

Log  A 
9.541. 

LogB 
0.242. 

Tab.  XXX.  B. 

Sun's 
True 
Azimuth. 

Sun's 
Com  pass- 
Azimuth. 

Comp. 
Error. 

Comp. 
Dev. 

TanX. 

TanT. 

Ang.X. 

Ang.  T. 

h    in 

o 

o 

o 

o 

5  49 

O.O2I 

9.562 

0.263 

20.05 

61.4 

N  81.4  W 

N64  W 

17.4  W 

8.6  E 

56 

008 

549 

250 

19-5 

60.65 

80.  i 

72 

8.1 

17.9 

6      2 

9.996 

537 

238 

19.0 

59-95 

79.0 

75 

4.0 

22.O 

7 

987 

528 

229 

18.6 

59-45 

78.1 

72 

6.1 

19.9 

12 

977 

518 

219 

18.25 

58.85         77.1 

67 

IO.I 

15-9 

6  18 

9.966 

9-507 

0.208 

17.8 

58.2 

76.0 

60 

16.0           10.0 

25 

953 

494 

195 

17-3 

57-45 

74-7 

51 

23-7 

2.3  E 

30 

943 

484 

185 

16.95 

56.8s 

73-8 

43 

30.8 

4.8  W 

34 

935 

476 

177 

16-7 

56.35 

38 

35-1 

Q    I 

44 

916 

457 

158 

16.0 

55-2 

71.2 

28 

43-2 

17.2 

6  49 

9-9 

06 

9-447 

0.148 

15.6 

S4-6 

70.2 

23 

47.2        '21.2 

55 

»5 

436 

137 

15-3 

53-9 

69.2                      21 

48.1        :  22.1 

59 

8 

428 

129 

15.0 

53-4    i      68.4               23 

45-4 

19.4 

7    5 

87S 

416 

117 

14.6 

52.6          67.2 

28 

39-2 

13.2 

12 

861 

402 

103 

14.2 

51-7  !    65.9 

36 

29.9 

3-9  W 

Ex.  3.—  1875, 

Nov.  20  :  At  sea 

,  in  Lat.  3°  58'  N,  Long.  14°  3'  W,  about  6h  45in  A.  M., 

Ship  A.  T.,  weather  threatening, 

made  the  following  observations,  on  a  partial  swing, 

to  cover 

probable  courses  of  the 

Ship,  during  two  or  three  days  : 

Re'cord 

of  Observations.                Watch  2m  ios  slow 

on  Ship  A.  T. 

Stand.  Compass. 

Ship  Mid.  A.  T.  Nov.    19*  i8h.8 

Watcb- 
Time. 

Corrected 
A.T. 

Long,  in  W 
Greenwich  date 

+      o  .9 
19    19  .7 

Ship's 
Head. 

Sun's 
Centre. 

h   m 

s 

h 

m 

o 

O's  Dec.  S  19^.7 

6  38 

20 

6  40.5 

South. 

S  5i-5  E 

44 
48 

55 

0 

50 

IO 

46.2 

51.0 

57-3 

S    15°  E 

29 
46 

49-o 
46.0 
42-5 

Pol.  Dist.  K>9°.7<  Log  A  9.316  )    Tab. 
Co-Lat.       86  .0  (  Log  B  0.855  ]  XXX 

59 

0 

7 

1.2 

60 

38.0 

Mag.  Var.  190.3  W    (Tab.  LVI) 

Tabular  Form  by  Azimuth-Tables. 

Sun's 
H.  Angles 

Tab. 
1  XXX.A. 

Log  A     Log  B 
9.310.     O.H55. 

Tab.  XXX.  E. 

O's  True 
Azimuth. 

Compass- 
Azimuth. 

Compass 
Error. 

•  Compass- 
l)cv. 

TanX.    TanT. 

Ang.X 

Ang.T. 

h     m 

5  i9-5 

0.077 

9-393     °-932 

0 

13.9 

83°3 

N  uoC.6  E 

N  i28°.5  E 

I7?9W 

0 

1.4  E 

• 

13-8 

088 

404        943 

[4.2 

83.5 

110.7 

131.0 

20.3 

i.o  W 

8.0 

099 

415        954 

14.6 

83-7 

110.9              134.0 

23.1 

3.8  W 

2-7 

no 

426        965 

[4.9 

83.8 

1  1  1.  1              137.5 

26.4 

7.1 

458-8 

117 

433        972 

15.2 

83-9 

111.3              142.0 

30-7 

11.4 

METHOD    OF    TIME-AZIMUTHS. 


31 


JEx.  4. — 1875,  July  16:  At  sea,  in  Lat.  o°  31'  S,  Long.  20°  15'  W,  about  nh  30™  P.  M., 
made  the  following  observations  of  the  Moon,  on  a  partial  swing  of  the  Ship,  for  Com- 
pass-Deviations : 


Observations  and  Deduced  Data. 


Standard  Compass. 

R.A.M.0 

7'»  31  '".S 

R.  A.  Moon. 

Watch- 

Ship 

MJMIII 

Green- 

Time. 

Ship's 
He«d. 

Moon's 
Centre. 

Time. 

date. 

R.  A.  Mer. 

At  G.  I2h 

T.LXV 
Corr. 

Reduced 
R.A.  Q 

h    in      s 

o 

h    m 

h    in 

h    m 

h    m 

in 

h     m 

II    14  20 

ssw 

Si6.oW 

II    2O.O 

12  41 

18  57.8 

18  27.5 

+1.6 

18  29.1 

17    o 

SbyW 

17.0 

22.7 

44 

19    0.5 

27-5 

1.7 

29.2 

19  30 

South 

18.5 

25.2 

46 

3-° 

27-5 

1.8 

29-3 

23   2O 

SbvE 

21.5 

29.0 

5° 

6.8 

27-5 

1.9 

29-4 

27    o 

SSE 

25-5 

32.7 

54 

10.5 

27-5 

2.1 

29.6 

3°    ° 

SEbyS 

29.0 

35-7 

57 

13-5 

27-5 

2.2 

29-7 

35  40 

SE 

33-o 

41.4 

13    2 

19.2 

29.8 

O.I 

29.9 

W.  Time  slow  5™.  7  on  Ship  M.  T. 

Mid.  M.T.July  16*11^.5 

Long.  W  +      i  .3 

Mid.  Greenwich  date  16    12  .8 


R.  A.  Mean  Sun  7h  3 7™. 8 


Long,  in  ih  21™  W 
(['s  Dec.  S  28°  if 

Pol.  Dist.  6i°-7  (  Log  A  9.398  )     Tab. 
Co-Lat.     89  .5  I  Log  B  0.588  5  XXX 

Mag.  Var.  I9°.2  W    (Tab.LVI) 


Tabular  Form  by  Azimuth-Tables. 


Moon's 

Log  A 

9  39S 

Log  B 

0  5M8 

Table  XXX.  B. 

Hour- 
Angles 

Table 
XXX.  A. 

G  's  True 
Azlmutn. 

Compass 
Azimuth. 

Compass 
Error. 

Compass- 
l)ev. 

West. 

TanX. 

TanT. 

Ang.  X. 

Ang.  Y. 

h    m 

o  28.7 

1.203 

0.601 

1.791 

0 

75-9 

89.1 

0 

S  i6.oW 

2.8  W 

I6.4E 

31-3 

165 

563 

753 

74-7 

89.0 

14-3 

17.0 

2-7 

16.5 

33-7 

133 

721 

73-6 

88.9 

15-3 

18.5 

3-2 

16.0 

37-4 

089 

485 

675 

71.9 

88.8 

16.9 

21-5 

4.6 

14.6 

40.9. 

048 

446 

636 

70.3 

88.7 

18.4 

25-5 

7-1 

12.  1 

43-8 

1.018 

416 

606 

69.0 

88.6 

19.6 

29.0 

9-4 

9-8 

.  49-3 

0.967 

365 

555 

66.6 

88.4 

21.8 

33-° 

II.  2 

o.O 

In  this  Example  for  a  partial  swing,  the  Moon  being  taken  when  her 
change  of  Declination  was  comparatively  slow,  the  results  are  as  relia- 
ble as  they  would  have  been  if  obtained  from  observations  of  the  Sun 
under  similar  circumstances.  The  additional  labor  of  the  working-out 
is  trifling,  as  it  only  involves  the  six  extra  columns  in  the  first  Table? 
which  are  formed  easily  and  rapidly.  The  Hour- Angles  are  very  small. 

Such  an  operation,  although  performed  at  an  unusual  hour  of  the 
day,  might  be  the  means  of  anticipating  serious  embarrassment,  espe- 
cially if  threatened  with  thick  weather,  and  actually  deprived  of  obser- 
vations for  two  or  three  days. 


32  FINDING   THE    COMPASS-ERROR. 


C.  METHOD  OF  CIRCUMPOLAR  AZIMUTHS. 

36.  The  Circumpolar  Azimuth  a  Modified  Form  of 
the  Time-Azimuth. — Circuinpolar  Azimuths,  as  the  name  implies, 
are  applicable  solely  to  Circumpolar  Stars ;  that  is,  to  those  which  never 
set  in  the  horizon  of  the  observer,  but  are  always  found  above  it.    The 
method  is  a  modified  form  of  the  Time- Azimuth,  receiving  certain  special 
simplifications,  which  are  the  more  conspicuous  and  practically  useful  in 
proportion  as  the  Polar  Distance  of  the  Star  is  less,  or  in  proportion  to  its 
nearness  at  all  times  to  the  Elevated  Pole.    This  will  be  evident,  since,  as 
all  the  Fixed  Stars  require  sensibly  the  same  time  (nearly  24h)  in  making 
their  apparent  revolutions  about  the  Earth,  or  its  axis,  supposed  indefi- 
nitely produced,  it  follows  that  the  nearer  they  are  to  that  axis,  and  the 
smaller  the  circle  described  by  them,  the  slower  their  apparent  daily 
motions;  so  that,  when  quite  near  the  Pole,  they  appear  almost  entirely 
at  rest,  in  comparison  with  those  Stars  which  are  much  farther  from  it. 

Polaris,  or  the  "Pole  Star,"  is  a  notable  example  of  this  condition 
of  things  to  observers  in  the  Northern  Hemisphere;  for,  its  Polar  Dis- 
tance being  at  present  less  than  i^°,  its  greatest  change  in  Azimuth 
from  east  to  west  and  from  west  to  east,  to  an  observer  so  far  north 
as  in  the  Latitude  of  60°,  is  less  than  5°.$  during  I211  of  time,  while  to 
an  observer  at  the  Equator  it  is  even  less  than  3°.o  during  the  same 
time.  So  well  is  this  apparent  fixedness  of  the  Pole-Star  understood  by 
Navigators,  that  not  a  few  have  regarded  it  as  identical  with  the 
North  Pole  itself,  and,  in  sighting  it  with  their  compasses,  have  deduced 
their  Compass-Errors  therefrom  as  if  it  were  actually  at  the  North  Pole. 
But  such  an  assumption  may  involve,  though  not  necessarily  so,  a  quarter 
to  nearly  half  a  Point  of  Error;  that  is  to  say.  ±  3°.o  to  ±  5°.5,  accord- 
ing to  the  time  and  place  of  observation  within  the  range  of  Latitudes 
indicated. 

Now  the  motion,  in  Azimuth,  of  a  Circumpolar  Star,  having  a  small  Pol. 
Dist.,  is  not  only  very  slow,  but  really  quite  variable ;  being  slowest 
at  or  near  its  greatest  Elongations,  east  or  west,  and  quickest  when  on 
or  near  the  Meridian,  above  or  below  the  Pole.  Hence,  in  assuming  it 
to  be  practicable  to  get  reliable  Azimuths  of  such  a  Star  with  the  compass, 
it  is  desirable  to  know  when  to  observe,  in  order  to  find  the  Star  at  its 
greatest  Elongation,  which  is  the  most  favorable  time,  or,  losing  that,  to 
have  its  True  Azimuth  at  any  intermediate  time  of  observation. 

37.  Tahle  or  Circumpolar  Azimuths :  Description  of 
Tah.  XL<  Tor  Polaris.— Such  is  the  object  of  Tab.  XL,  which  is 
adapted  to  Polaris,  and  comprises  the  True  Azimuths  of  this  Star,  at 
intervals  of  every  hour  in  Sid.  Time,  and  for  all  Latitudes  from  the 
Equator  to  60°  north.     Thus,  for  any  stated  Lat.,  the  True  Azimuth  of 
the  Star  is  given — 


'   J^ 


METHOD    OF    CIRCUMPOLAR    AZIMUTHS. 


First,  at  its  greatest  Western  and  Eastern  Elongations;  being  set  down 
in  Cols.  Ill  and  XV,  adjacent  to  the  corresponding  Sid.  Time.  As,  for 
example,  in  Lat.  30°,  at  the  greatest  Western  Elongation,  the  True  Az. 
is  K  i°.6  W  at  y11  9m-9  Sid.  T.,  while  at  the  greatest  Eastern  Elongation 
the  True  Az.  is  X  i°.6  E  at  iQ11  i6m.i  Sid.  T. 

Secondly,  opposite  to  the  Latitudes  in  •  the  intermediate  columns, 
IV-XIV,  the  Sid.  Times  of  which  being  placed  at  top  and  bottom.  The 
Sid.  Times  at  top,  commencing  with  the  middle  column  at  ih  i3m,  when 
the  Star  is  on  the  Meridian  above  the  Pole,  increase  by  hourly  intervals, 
as  the  Star  moves  westwardly  from  the  Meridian  to  its  greatest  W.  E., 
when  the  Sid.  Times  are  adjacent  to  the  Latitudes;  after  which,  during 
its  return  eastwardly  towards  the  Meridian,  the  Sid.  Times  are  found  at 
bottom,  till  at  13*  13™  the  Star  is  again  on  the  Meridian  below  the  Pole. 
Then,  as  the  Star  continues  its  eastwardly  movement  below  the  Pole 
again  moving  from  the  Meridian,  the  Sid.  Times  are  still  found  at  bottom, 
till  it  arrives  at  its  greatest  E.  E.,  when  the  Sid.  Times  are  adjacent  to 
the  Latitudes;  after  which,  during  its  return  westwardly  towards  the 
Meridian,  the  Sid.  Times  are  found  at  top  till  it  finally  arrives  at  the 
point  of  departure  on  the  Meridian  above  the  Pole,  when  the  Sid.  T.  is 
again  ih  13™.  This  Table  is  computed  for  January  i,  1875,  but  will  not 
be  materially  at  fault  for  at  least  ten  years. 

38.  Two  Distinct  Problems  in  the  Use  of  Tab.  XL.— 

Although  the  practical  object  in  the  use  of  Tab.  XL,  so  far  as  the  com- 
pass is  concerned,  is  to  get  the  True  Azimuth  of  the  Star,  there  are  two 
distinct  problems  to  be  solved  in  connection  therewith. 

First,wiih  a  tolerable  Lat.  and  Ship-Time,  having  observed  a  Compass- 
Azimuth  of  Polaris  and  noted  the  Ship  T.  thereof,  it  is  required  to  take 
out  the  corresponding  True  Azimuth  of  the  Star,  and  thus  to  obtain  the 
Error  of  the  Compass;  or, 

Secondly,  with  a  very  uncertain  Lat.  and  Ship-Time,  it  is  required  to 
find  the  Ship  T.  of  the  Star's  greatest  Western  or  Eastern  Elongation, 
whichever  may  occur  during  the  night,  to  observe  a  Compass- Azimuth 
at  that  Time,  and  to  take  out  the  corresponding  True  Azimuth,  in  order 
to  obtain  a  reliable  Compass-Error,  independently  of  the  uncertainties 
of  the  Data. 

39.  Rule:  To  find  the  True  Azimuth' of  Polaris  by 
Tab.  XL. — Thus,  we  shall  have  the  following  liule: 

a)  To  take  out  the  True  Azimuth  for  any  Ship- Time. — Find  the  Green- 
wich date  for  the  Ship  T.  to  the  nearest  tenth  of  the  hour,  in  the  usual 
manner ;  then,  according  as  the  T.  is  A.  or  M.,  take  out  the  K.  A.  of  the 
True  or  Mean  Sun  from  Tab.  LIX,  correcting  it  for  the  G.  date  by  Tab. 
LX1V  or  LXVI,  and  add  it  to  the  Ship  T.  (rejecting  24^  if  the  sum  be 
greater  than  24^),  and  the  result  will  be  the  Ship  Sid.  Time.1  With  this 
Time  and  the  Lat.  enter  Tab.  XL  and  take  out  the  corresponding  True 
Azimuth,  as  required. 

1  Which  is  the  same  thing  as  the  E.  A.  of  the  Meridian  of  the  preceding  Articles. 


34 


FINDING    THE    COMPASS-ERROR. 


b)  To  find  the  Ship- Time  of  the  greatest  W.  or  E.  Elongation,  and  to  take 
out  the  corresponding  True  Azimuth. — Find  the  Greenwich  date  for  the 
supposed  Time  at  Ship.     With  this  date  take  out  from  Tab.  LIX  the  R. 
A.  of  the  True  or  Mean  San,  according  as  the  Time  is  A.  or  M.     Then, 
enter  Tab.  XL  with  the  supposed  Lat.,  and  subtract  the  R.  A.  before  men- 
tioned from  the  corresponding  Sid.  T.  (increased  by  241'  if  necessary)  of 
that  greatest  Elongation,  Eastern  or  Western,  which  gives  a  remainder 
as  Ship-Time  appropriate  to  the  night,  and  take  out  the  corresponding 
True  Azimuth.     Also,  make  the  Compass-Observation  at  that  Time. 

c)  Mark  the  Azimuth  N  according  to  the  Lat.  and  E  or  W  according 
as  the  Star  is  making  its  Eastern  or  Western  Elongation. 

4O.  Examples  of  Finding  Compass-Error  by  Circum* 
polar  Azimuths. 


Ex.  1.— 1875,  March  15  :  At  sea,  in  Lat, 
50°  1 1'  N,  Long.  45°  W,  about  9h  2oin  P.  M., 
Ship  A.  T.,  the  observed  bearing  of  Polaris 
was  N  I4°.5  E  :  Required,  the  Compass- 
Error. 

Ship  A.  T.  Mar.      isd    9b-3 
Long.  W  +3-o 

Greenwich  date     15    12  .3 

Ship  A.  T.  gh2om.o 

R.  A. True©    23  41    .7 
Ship  Sid.  T.       9     i    -7 

True  Az.    N     i°.8  W  (Tab.  XL) 
Comp  Az.    N  14  .5  E 
COMP.  ER.      16  .3  W 

Ex.  2. — 1875,  June  25:  At  sea,  in  Lat. 
42°  N,  Long.  155°  E,  at  nh  50™  P.  Mv  Ship 
M.  T.,  the  observed  bearing  of  Polaris  was 
N  27°  W:  Required,  the  Compass-Error 
and  Deviation. 

ShipM.  T.June     25**  nh.8 
Long.  E  -    10  .3 

Greenwich  date      25      1.5 

ShipM.  T.,  iih5om.o 
R.  A.  M.©  6  13  .2 
Ship  Sid.  T.  18  3  .2 

TrueAz.     N     i°.8  E  (Tab.  XL) 

Comp.  Az.  N  27  .o  W 

COMP.  ER.      28  .8  E 

Mag.  Var.        17  .5  E  (Tab.  LVI) 

COMP.  DEV.    1 1  .3  E 


Ex.  3« — 1875,  February  19:  At  sea,  in 
thick  weather,  blowing  a  gale  for  two 
days,  but  now  clearing  to  the  northward, 
and  Polaris  visible,  obtained  an  observa- 
tion for  Compass-Error  of  N  11°  W. 

Lat.    20°  N    \  Supposed,  with  possible 
Long.  30°  W  )     error  of  ±5°  each. 

About  7h  P.  M.,  Ship  A.  T. 

Ship  A.  T.  Feb.       i9d  7h 
Long  W  -f-    2 

Greenwich  date      19  9 


Ship  A.  T. 
R.  A.  True  0 
Ship  Sid.  T. 

True  Az.      N 


W 


Comp.  Az.  N  ii  .o  W 
COMP.  ER.          9  .7  E 

And  this  is  true  within*  ±o°.i,  even  with 
an  error  of  ±  10°  in  Lat.  and  ±  5°  in  Long, 
of  the  Ship's  place. 

Ex.  4. — 1875,  October  17:  At  sea,  in 
Lat.  52°  57'  N,  Long.  26°  19'  W,  about  7" 


P.  M.,  ship  A.  T. :  Required,  the  Ship  T.  of 
the  next  greatest  Elongation  of  Polaris. 

Ship  A.  T.  Oct.       I7d  7h.o 
Long.  W  -f    i  .8 

Greenwich  date     17  8  .8 

Sid.  T.  W.  E.    7h    8m 
R.  A.  True  ©  13   29 
Ship  A.  T.        17  39 

Ex.  5.— 1875,  May  31  :  At  sea,  in  Lat.  49°  15'  N,  Long.  43°  50'  W,  about  i2h  P.  M. 
approaching  the  American  coast,  and  anticipating  thick  weather,  desired  to  get  a  set 
of  Azimuths  for  Compass-Deviations,  sufficient  to  cover  the  probable  courses  of  the 
ship  for  the  next  day. 

Turning  to  Tab.  LIX,  the  R.  A.  True  Sun  is  seen  to  be  4h  32™  at  Greenwich  noon  ; 
next,  opening  at  Tab.  XL  and  comparing  this  R.  A.  with  the  Sid.  Times  of  the  Elon- 


METHOD    OF   ALTITUDE-AZIMUTHS. 


35 


gatioiis  of  Polaris,  it  is  seen  that  the  next  (Eastern)  Elongation  will  take  place  at 
i4b  47m  Ship  A.  T.,  or  at  2h  47™  hence.  Accordingly,  preparations  are  immediately 
made  to  take  the  observations  with  the  following  results. 

Mid.  Greenwich  date,  May  3id  I5h.6  ;  E.  A.  True  Sun,  4b  34m.6. 


Ship  A.  T. 

Ships  Head. 

Compass- 
Azimuth. 

Ship  Sid.  TJ  AJ2J|V 

1 

Compass 
Error. 

Mag.  Var. 

Compass 
Deviation. 

h    m 
12  30 

35 
39 
45 
50 
54 

WXW 

WbyN 
West 
W  by  S 
WSW 
SW  by  W 

N  26.5  E 
23.0 
20.5 
18.5 
17.0 
18.0 

h    m     j           o 
17  35        N  1.9  E 
40            1.9 
44             i-9 
5°             i-9 
55            2,0 
59            2.0 

24.6  W 

21.  1 

18.6 
16.6 
15.0 
16.0 

36.0  W 
36.0 
36.0 
36.0  ' 
36.0 
36.0 

i  ME 

14.9 
17.4 
19.4 

21.0 
2O.O 

D.    METHOD  OF  ALTITUDE-AZIMUTHS. 


41.  Fundamental    Principles  of  the  Altitude- Azi- 
muth.— In  an  Altitude- Azimuth   the   bearing  of  the   Sun  or  other 
heavenly  body  is  observed  with  the  compass,  at  the  same  time  that  its 
Altitude  is  measured  with  a  sextant,  octant,  or  other  reflecting  instru- 
ment. 

The  True  Azimuth  is  found  by  a  solution  of  the  Triangle  of  Position. 
The  Data  required  are — 

The  True  Altitude,  deduced  by  correcting  the  Observed  Altitude; 
The  Declination,  taken  from  the  Nautical  Almanac  or  Naut.  Tables; 
The  Latitude  in,  as  brought  up  by  the  .Reckoning  or  known  from 
previous  observation. 

42.  Rule :  To  make  the  Observation  of  an  Altitude- 
Azimuth. — Take  a  bearing  of  the  object  with  the  standard  compass, 
or  preferably  a  set  of  two  or  three  bearings  as  quickly  as  possible; 
bisecting  it  each  time,  if  it  have  a  sensible  disk  (Sun  or  Moon),  and 
noting  the  times  with  a  watch ;  also,  simultaneously,  with  each  bearing, 
take  the  Altitude  of  the  body  with  a  sextant  or  octant. 

Note  the  Heading  of  the  ship  with  the  same  compass,  and  the 
corresponding  Headings  with  the  steering-compasses;  also,  if  on  board 
an  iron  ship,  the  Angle  of  Heel,  and  whether  to  starboard  or  port. 

The  observed  bearing,  or  the  mean  if  several  be  taken,  is  the  required 
Compass-Azimuth ;  and  the  single  Altitude,  or  the  Mean  if  several  be 
taken,  is  the  observed  Altitude. 

43.  Rule:  To  find  the  True  Azimuth  hy  Computa- 
tion.— First,  prepare  the  Data,  as  follows: 

a)  Find  the  Greenwich  date  for  the  Ship  T.  of  observation,  either  by 
applying  the  Longitude  to  the  Time,  or  deducing  it  from  the  chronome- 
ter. 

b)  With  this  date,  take  out  the  Declination  of  the  body,  and  in  the 
case  of  the  Moon  the  Semi-Diameter  and  Horizontal  Parallax. 

c)  Get  the  True  Altitude  of  the  centre  by  applying  the  proper  cor- 
rections to  the  observed  Altitude. 


36  FINDING   THE    COMPASS-ERROR. 


Next,  compute  the  True  Azimuth,  as  follows  : 

d)  Add  together  the  True  Altitude,  the  Lat.,  and  the  Pol.  Dist.,  and 
take  the  Difference  between  their  Half-Sum  and  the  Pol.  Dist.  Then, 
using:  Tab.  IX  or  Tables  X  and  XI,  add  together  the  secant  of  the  Lat., 
the  secant  of  the  Alt.,  the  cosine  of  the  ^  Sum,  and  the  cosine  of  the 
Difference.  The  Half-Sum  of  these  four  logarithms,  rejecting  tens  from 
the  indices,  will  be  the  cosine  of  half  the  True  Azimuth,  which  take  out 
and  double,  and  the  result  will  be  the  required  True  Azimuth.  Mark  it 
N  or  S  according  to  the  Lat.,  and  E  or  W  according  as  the  Altitude  is 
increasing  or  decreasing,  or  as  the  object  is  East  or  West  of  the  Meridian. 

44.  Rule:  To  find  the  True  Azimuth  by  Aziiiiuth- 
Tables. — The  preceding  Kule,  by  Computation,  is  that  usually  given. 
But  the  process  of  getting  out  a  True  Azimuth  may  be  greatly  abridged, 
with  results  sufficiently  exact  for  practical  purposes,  by  special  Azimuth- 
Tables. 

Such  is  Table  XLI,  which  has  been  extended  and  otherwise  modified 
from  that  originally  given  by  another  hand.1  It  is  adapted  to  all  Lati- 
tudes from  the  Equator  to  80°  north  or  south;  to  all  Declinations  from 
o°  to  30°  of  the  same  or  contrary  name ;  and  to  all  Altitudes  from  o°  to 
82°. 

This  Table  consists  of  three  parts,  to  use  which,  after  preparing  the 
Data  as  before  described  (43),  proceed  as  follows  : 

With  the  Lat.  and  Alt.,  both  to  the  nearest. degree,  enter  Tab.  XLI, 
Part  I,  and  take  out  the  corresponding  tabular  number.  .  With  the  Dec. 
to  the  nearest  degree,  and  the  Diff.  of  the  Lat.  and  Alt.,  enter  Part  If, 
and  take  out  the  corresponding  number.  Then,  with  the  Sum  of  these 
two  tabular  numbers,  enter  Part  III,  and  take  out  the  corresponding 
True  Azimuth,  which  mark  in  the  manner  already  described  in  the  pre- 
ceding Rule. 

Remark  I. — The  arguments  of  the  Table  being  set  down  for 
every  even  second  degree,  it  will  be  necessary  to  take  Means  of  the  tab- 
ular numbers  in  Parts  I  and  II,  adjacent  or  cross,  according  as  one  or 
both  of  the  given  arguments  are  odd. 

In  general,  the  Azimuth  will  be  obtained  in  this  manner  within  a  half- 
degree,  or  ±o°.5,  of  that  found  by  computation.  But,  whenever  deemed 
expedient  to  take  the  Data  more  closely,  it  is  easy  to  proportion  for  a 
more  precise  result. 

Remark  II :  Table  of  Direct  and  Limiting  Value* 
of  the  Altitude-Azimuth. — Table  XLII  will  be  found  useful,  in 
furnishing  a  convenient  check  against  any  serious  error  in  getting  out 
an  Altitude- Azimuth,  whether  by  computation  or  by  Table  XLI. 

1  Azimuth  Table  ;  to  facilitate  the  process  of  finding  the  True  Bearing  of  the  Sun,  etc.,  by 
A.  .C.  JOIIXSONT,  R.  N.,  London,  1867. — This  is  contained  in  a  small  pamphlet  by  th#t 
author  under  the  title  How  to  find  the  Lvnyilude  simultaneously  with  the  Latitude  at 
Noon,  etc. 

There  is  also  an  extensive  set  of  Tables  for  finding  the  Altitude-Azimuth  directly, 
consisting  of  365  pages,  4to,  by  THOMAS  LYNN,  London,  1829. 


METHOD    OF    ALTITUDE-AZIMUTHS. 


37 


45.  Examples  of  Altitude-  Azimuths. 

Ex.  1.—  1868,  August  5  :  At  sea,  in  Lat. 

By  Azimuth-Tables: 

23°  47'  N,  Long.  110°  15'  W,  at  4h  6m  P.  M., 

Lat.  27°.3Alt.   22°.5       (Tub.  XLI)  43 

Ship  A.  T.,  all  by  account,  the  observed 

Diff.     4  .8  Dec.  10  .4                            884.6 

bearing  of  the  Sim's  centre  was  S  85°  W  and 

TrueAz.        Sii5.8W             Sum  927.6 

its  True  Alt.  was  30°  26^.4  :  Required,  the 

Comp.  Az.      S  105  .0  W 

Compass-Error. 

COMP.  ER.           10  .8  E 

Ship  T.     5d    4b.i 

Long,  iu  -f-    7  .3 

Ex.  3.—  -1871,  August  16:  At  anchor, 

Gr  date   5114                        <  Disco   Island>  in  Lat-  69°  *4'  N,  Long.  53° 

!  1  8'  W.    With  the  Data  of  Ex.  i  in  Time- 

Q'sDec.            N    1  6°  51'  |  —  o'.7 

Azimuths  (29),  and,  in  addition,  the  True 

Rednforiih4        —      8          7.7 

Altitude  of  the  Sun's  centre  21°  24'  :     Re- 

Red. Dec.               16   43            .3 

quired,  the  Compass-Error. 

Pol.  Dist.                73    17 

By  Computation  : 

By  Computation  : 

Pol.  Dist.     76°  17' 

Pol.  Dist.    73°i7/ 

Lat.  in          69    14     sec    0.4503 

Lat.  in         23   47      sec    0.0385 

True  Alt.      21    24      sec     0.0310 

True  Alt.    30  26     sec    0.0644 

Sum             166    55 

Sum            127  30                                                 y2  slim          83    27^  cos    9.0566 

^  Sum         63   45     cos    9.6457 

Diff.                7    io)4  cos    9.9966 

Diff.               9  32     cos    9.9940 

J9-5345 

19.7426 

y2  Az.            54    12     cos    9.7672 

yz  Az.           41    58     cos    9.8713 

TrueAz.NioS   24     W 

True  Az.  N  83   56  W 

By  Azimuth-Tables  : 

By  Azimuth-Tables  : 

Lat.  69°.2    Alt.  2i°.4(Tab.  XLI}   240 

Lat.  24°    Alt.   300.0    (Tab.  X  LI)    51 

Diff.  47  .8    Dec.  13.7                       668 

Diff.    6    Dec.  17.0                          772              TrueAz.        N  108  .3  W        Sum  908 

True  Az.       N  83  .5  W           Sum  823              Comp.  Az.     N    36  .0  W 

Comp.  Az.    N  95  .0  W                                       COMP.  ER.           52  .3  W 

COMP.  ER.        1  1  .5  E 

Ex.  4.—  1868,  December  24:  At  sea,  in 

Ex.  2.—  1871,  April  17:  At  sea,  in  Lat.     Lat.  41°  57'  N,  Long.  20°   19'  W,  with  a 

27°  19'  S,Long.  75°  52'  E,  the  Sun's  bear-     Time-Sight  of  the  Moon. 

ing  by  compass  was  N  75°  W,  at  3h  51™  P. 
M.,  Ship  A.  T.,  and  the  True  Alt.  at  the 
same  time  was  22°  29'  :  What  was  the  Com- 

Compass-Bear. S  40°  o'  E  )  At  4h         p  M 
True  Altitude       37   22      >        Shin  M.'  V 

pass-Error  ? 

Dec.                        7   37     3 

ShipT.            i7d    3*1.8 

By  Computation  : 

Long,  in           —     5  .0 

Pol.  Dist.     82°  23' 

Gr.  date  Apr.  16  22  .8 

Lat.  in         41    57     sec    0.1286 

17—1  .2 

True  Alt.    37    22     sec    0.0998 

O's  Dec.         N  10°  25'  |  -f  o'.9 

Sum            161    42 

Rednforih.2      —     i          i  ^o                          X  Sum        80    51      cos    9.2015 

Red.  Dec.            10  24                                       Diff.               i    32     cos    9.9998 

Pol.  Dist.           100  24                                                                                 I9-4297 

By  Computation:                                                           #  Az'          5846     cos    9.7148 
TrueAz.Nii7    32  E 
Pol.  Dist.  100°  24' 

Lat.  in         27    19     sec    0.0514             !  By  Inspection  :                                                     . 

True  Alt.    22    29     sec    0.0343                    Lat.  42°     Alt.  37°.o  (Tab.  XLI)    113 

Sum           150    12 

Diff.     5      Dec.    7  .5                         818 

l/2  Sum         75      6     cos    9.4102 

TrueAz.      N  117  .3  E          Sum  931  • 

Diff.              25    18     cos    99562 

Comp.  Az.    N  140.0  E 

T  c\  A  £*j  i 

COMP.  ER.          22  .7  W 

1y-45zl 

y2  Az.          57    21      cos    9.7260 

Tr.  Az.   S  115    42      W 

38  FINDING    THE    COMPASS-ERROR. 


E.  METHOD  OF  TIME- ALTITUDE-AZIMUTHS. 

46.  Fundamental    Characteristics. — The  Time-Altitude- 
Azimuth  has  for  its  Data  the  Hour- Angle,  Altitude,  aud  Declination ;  and 
as  these  comprise  a  part  of  the  Data  of  the  Time  and  Altitude  Azimuth, 
which  have  already  been  fully  considered  when  treating  the  last-named 
methods,  they  need  no  further  explanation  in  this  place. 

The  solution  of  the  Triangle  of  Position  for  the  Azimuth,  with  these 
Data,  is  at  first  sight  more  simple  than  for  either  the  Time  or  Altitude 
Azimuth,  inasmuch  as  it  requires  only  the  addition  of  three  logarithms 
directly  from  the  Data,  the  sum  of  which  being  the  logarithm  of  the 
Azimuth.  But,  unfortunately,  it  has  the  very  serious  disadvantage  of 
giving  an  ambiguous  result,  which,  although  removable  in  certain 
special  cases,  is  not  in  general  freed  from  the  ambiguity,  even  with  con- 
siderable complexity  of  cases.  Moreover,  as  a  method  for  finding  Serial 
Compass-Errors,  its  use  is  attended  with  the  further  disadvantage  that, 
as  the  H.  A.  and  Alt.  are  both  changeable  from  one  observation  to 
another  of  the  series,  there  is  really  more  labor  required  to  get  out  a 
set  of  True  Azimuths  by  this  method  than  by  either  of  the  preceding. 

On  the  whole,  therefore,  if  to  what  has  already  been  said  it  be  added 
that  the  Data  for  the  solution  of  the  Time- Azimuth  are  generally  availa- 
ble whenever  those  for  the  Time- Alt.  Azimuth  are,  there  would  seem  to 
be  little,  if  any,  occasion  to  resort  to  the  use  of  the  latter  method  in 
preference  to  the  former. 

Nevertheless,  the  following  Kule  for  its  solution  is  given,  together 
with  a  few  Examples  to  illustrate  the  application,  so  that  it  may  be  con- 
veniently in  hand  for  any  use  that  may  be  made  of  it. 

47.  Rule:  To  find  the  True  Azimuth  by  Computa- 
tion.— The  Data  being  given  or  prepared  by  the  preceding  Rules,  pro- 
ceed as  follows : 

a)  Using  Tab.  IX :  To  the  sine  of  the  H.  A.  add  the  secant  of  the 
Alt.  and  the  cosine  of  the  Dec. ;  their  Sum,  rejecting  10  from  the  index, 
is  the  sine  of  the  Angle  Q,  which  take  out. 

There  will  now  be  two  general  Cases,  according  as  the  Pol.  Dist.  is 
greater  or  less  than  90°. 

b)  Case  I :   The  Pol.  Dist.  greater  than  90°. — Subtract  the  Aug.  Q 
from  1 80°,  and  the  result  will  be  the  True  Azimuth,  always  greater 
than  90°. 

c)  Case  II :  The  Pol.  Dist.  less  than  90°. — In  this  case  there  is  gen- 
erally a  doubt,  whether  the  Aug.  Q  or  its  Supplement  is  to  be  taken 
as  the  True  Azimuth,  with  the  following  exception  : 

Whenever  the  Alt.  is  less  than  the  Dec.,  the  Aug.  Q  will  be  the  True 
Azimuth,  less  than  90°. 

'  Otherwise,  whenever  the  Alt.  is  greater  than  the  Dec.  (a  very  com- 
mon condition),  the  result  will  be  in  doubt,  aud  reference  must  be  made 
with  the  Data  to  Tab.  XXXI,  where  it  will  generally  be  apparent,  at 
sight,  whether  the  True  Azimuth  is  really  less  or  greater  than  90°.  If 


METHOD    OF    TIME-ALTITUDE-AZIMUTHS. 


39 


less,  the  Ang.  Q  is  the  True  Azimuth ;  if  greater,  the  Supplement  by 
subtracting  from  180°  is  to  be  taken  as  the  True  Azimuth. 

In  many  instances  it  will  be  easy  to  decide  during  the  observation, 
whether  the  object  is  on  the  side  of  the  Prime  Vertical  towards  the 
Elevated  Pole,  or  the  reverse.  In  the  former  condition,  the  Ang.  Q  will 
be  taken  as  the  True  Azimuth ;  in  the  latter,  its  Supplement  will  be 
taken,  greater  than  90°. 

cl)  In  every  case,  mark  the  True  Azimuth  N  or  S  according  to  the 
Lat.,  and  E  or  W  according  as  the  H.  A.  is  E  or  Wof  the  Meridian. 

48.  Examples  of  Finding  the  True  Azimuth. 


Ex.  1. — At  sea,  in  Lat.  30°  N,  obtained 
the  following  Data  in  an  observation  of 
the  Sun  : 

H.  A.  West     4h  o™ 
T.  Alt.  I3°.6 

Dec.  S  20  .o 

Required,  the  True  Az. 
By  Computation : 
H.  A.        4h   o' 
Alt.  i3°.6 

20    .0 


sin  9.9375 
sec  0.0124 
cos  9.9730 
sin  9.9229 


Dec. 

Ang.  Q     56  .9 

or, 

TRUE  Az.  N  123°.  i  W 
(Since  this  falls  into  Case  I.) 
Ex.  2. — At  sea,  in  Lat.  9°  N",  obtained 
the  following  Data  in  an  observation  of 
the  Sun  : 

H.  A.  East    5b  42™ 
Alt.  7°  31' 

Dec.          N  22     o 
Required,  the  True  Az. 


H.  A.        5*  42 
Alt.  7°  3 1' 

Dec.        22°.o 
Ang.  Q  68  .8 
or, 

TRUE  Az.  N  68°.8  E 
(Since  this  falls  into  the  exception  of 
Case  II.) 


sin  9.9987 
sec  0.0037 
cos  9.9672 
sin  9.9696 


Ex.  3.— Given  the  Data  in  Lat  40°. i  N 
for  the  Son  : 

H.  A.  West  5h  6™.8 
Alt.  22°  53' 

Dec.        N  20°  .3 

Required,  the  True  Az. 

H.  A.  5h  6m.8  sin  9.9882 

Alt.  22°  53'  sec  0.0356 

Dec.  200 .3  cos  9.9722 

Ang.  Q  82  .2  sin  9.9960 

This  is  doubtful ;  but  a  reference  to 
Tab.  XXXI  shows  that  Ang.  Q  is  to  be 
taken ;  or, 

TRUE  Az.  N  82°.2  W 

Ex.  4. — With  the  same  Lat.  and  Dec., 
and — 

H.  A.    3h  56m.6 

Alt.      36°  is' 
Required,  the  True  Az. 

H.  A.        3b  56m.6     sin  9.9337 
Alt.  36°  15'        sec  0.0934 

Dec.          2o°.3          cos  9.9722 
Ang.  Q     86  .7          sin  9.9993 
Here,  again,  we  are  in  doubt,  but,  by 
Tab.  XXXI,  it  is  evident  that  we  must  use 
the  Supplement  of  Q  ;  or, 

TRUE  Az.  N  93°-3  W 


49.  Table  of  Time-Alt.  Azimuths.— Tab.  XLVIII  consists 
of  two  parts,  namely,  of  a  first  part,  of  which  the  arguments  are  the 
Azimuth  and  Declination,  and  of  a  second  part,  of  which  the  arguments 
are  either  the  Hour-Angle  and  Altitude,  or  the  Position-Angle  and  Lati- 
tude; that  is  to  say,  either  pair  of  arguments  at  the  same  time. 

This  Table,  although 'having  the  title  of  Time-Alt.  Azimuths,  is  not 
designed  exclusively  for  that  object,  but  to  answer  several  purposes,  as 
follows : 

1.  To  find  the  Time- Alt.  Azimuth,  having  the  H.  A.,  Alt,  and  Dec.  ; 

2.  To  find  the  Altitude,  having  the  Azimuth,  Dec.,  and  H.  A. ; 


40 


FINDING   THE    COMPASS-ERROR. 


3.  To  find  the  Hour-  Angle,  having  the  Azimuth,  Dec.,  and  Alt.;  and, 

4.  To  find  the  Position-Angle,  having  the  Azimuth,  Dec.,  and  Lati- 
tude. 

Whenever  at  least  two  of  the  Data,  one  of  which  for  each  part  of 
this  Table,  correspond  exactly  or  very  nearly  to  the  tabular  arguments, 
a  True  Azimuth,  or  either  of  the  other  quantities,  with  the  requisite 
Data,  may  be  taken  out  at  sight  with  sufficient  accuracy  for  any  ordi- 
nary requirement  ;  but,  if  the  Data  do  not  correspond,  and  interpolations 
are  necessary,  it  will  generally  be  easier  to  compute  the  Azimuth  by 
the  preceding  Rule.  For  an  off-hand,  rough  approximation,  the  Table 
is  convenient  for  the  several  purposes  for  which  it  is  designed. 


.  Examples  or  the  Use  of  Tali,  XLVOI. 


Ex.  1.— With  the  Data  of  Ex.  i  (48), 
what  is  the  True  Azimuth  ? 

With  H.  A.  4h  om  and  Alt.  i3°.6,  we  find 
for— 

Log  B    0.050 

And  with  this  and  Dec.  20°  in  first  part, 
we  find  for — 

True  Az.  i23°.o 

Ex.  2.— With  Az.  70°,  Dec.  10°,  and  H. 
A.  2h  20™,  what  is  the  True  Alt.  f 
Entering  first  part,  we  fiud  for — 

Log  A    0.020 
With  this  and  H.  A.  in  second  part,  we 


find  for- 


TrueAlt  53°.] 


Ex.  3.— With  Az.   151°,  Dec.  20°,  and 
Alt.  60°,  what  is  the  H.  A.  ? 

With  Az.  and  Dec.  in  first  part,  we  find 
for— 

Log  A    0.287 

With  this  and  Alt.  in  second  part,  we 
find  for— 

H.  A.    ih  om 

Ex.  4.— With  T.  Az.  21°,  Dec.  17°,  and 
Lat.  42°,  what  is  the  Pos.  Angle  ? 

With  Az.  and  Dec.  in  first  part,  we  find 
for— 

Log  A    0.426 

With  this  and  Lat.  in  second  part,  we 
find  for — 

Pos.  Ang.    i6°.2 


F.  TRANSITION- AZIMUTHS. 


51.  Fundamental  Principles. — The  method  of  Transition- 
Azimuths  is  based  on  the  relative  changes  of  the  Hour- Angle  and  Alti- 
tude of  a  celestial  object.  The  H.  A.  of  an  object,  whatever  its  Position, 
changes  uniformly ;  but  the  changes  of  the  Alt.  are  variable,  depend- 
ing on  the  Position  of  the  object  with  respect  to  the  observer's  Zenith. 
Accordingly,  whenever  two  or  more  observations  of  an  Altitude  are 
made  and  the  corresponding  Hour-Angles  noted,  it  will  be  easy  to  obtain 
a  characteristic  or  distinguishing  relation  of  these  two  elements,  by 
deducing  the  change  of  Alt.  in  a  unit  of  the  H.  A.  Interval;  and  this 
definite  relation,  in  a  given  case,  will  always  correspond  to  a  particular 
Azimuth  of  the  object ;  so  that  whenever  we  have  obtained  the  former, 
together  with  the  observer's  Latitude,  they  constitute  the  requisite  Data, 
by  means  of  which  we  may  readily  find  the  latter.  This  method  is 
rarely,  if  ever,  resorted  to  as  a  direct  means  of  getting)  the  True  Azi- 
muth 5  but  it  may  be  used,  often  with  advantage,  in  connection  with 


METHOD    OF    TRANSITION- AZIMUTHS.  41 


observations  for  the  Time  and  Position  near  the  Meridian,  when  the 
peculiar  Data  required  for  its  use  are  sometimes  obtained  ;  it  being  only 
necessary  that  a  Compass-Azimuth  of  the  object  be  taken  as  near  the 
middle  of  the  Time-Interval,  between  the  two  observations  of  Altitude, 
as  practicable.  With  the  True  Azimuth  and  Compass-  Azimuth  of  the 
object  known,  the  Compass-Error  is  found  by  the  usual  comparison. 


*2.  Preliminary  Remark  oil  the  Preparation  of  the 
Data.  —  The  Data  being  furnished  to  hand  by  the  direct  observation, 
from  which  the  True  Azimuth  is  merely  an  incidental  determination, 
there  is  no  necessity  to  enter  into  details  as  to  its  preparation.  It  will 
be  sufficient  to  remark  that,  the  Times  of  the  observed  Altitudes  being 
noted  with  a  watch  whose  error  on  Ship  Ap.  Time  is  known,  the  Diff. 
of  the  corrected  W.  Times  will  be  the  required  H.  A.  Interval,  when  the  Sun 
is  the  object  observed. 

Moreover,  as  the  W.  Time  Interval  in  these  cases  should  hardly  exceed 
a  half-hour,  the  small  Error  due  to  the  Difference  between  Ap.  and  M. 
Time  (only  seldom  so  much  as  o8.6  in  that  Interval)  may  be  disregarded, 
and  the  W.  T.  Interval  itself  used  as  the  H.  A.  Interval.  With  this  con- 
dition, tliis  method  is  entirely  independent,  even  with  a  large  uncer- 
tainty,1 of  the  Ship-Time. 

For  all  other  objects  than  the  Sun,  it  will  not  be  advisable  to  use  the 
W.  T.  Interval  for  the  proper  H.  A.  Interval,  as  the  difference  between 
them,  and  the  consequent  Error,  might  be  too  large  to  be  admitted  ;  but, 
in  such  cases,  the  H.  A.  Interval  should  be  deduced  by  taking  the  Diff. 
of  the  Hour  -Angles2  found  for  the  two  noted  W.  Times  of  observation  in 
the  usual  manner  (20,  b). 

53.  Rule  :  To  get  the  True  Azimuth.—  Having  the  Data 
prepared,  the  True  Azimuth  by  this  method  may  be  found  either  by 
Logarithmic  Computation  or  by  Tabular  Inspection. 

a]  By  Logarithmic  Computation.  —  Using  Tab.  IX  :  To  the  cosecant  of 
the  H.  A.  Interval  (or  W.  Time  Interval  for  the  Sun)  add  the  sine  of 
the  Alt.  Diff.  and  the  secant  of  the  Lat.;  their  Sum,  rejecting  tens  from 
their  indices,  will  be  the  sine  of  the  True  Azimuth,  which  take  out,  and 
mark  it  N  or  S  according  as  the  object  is  north  or  south  of  the  Prime 
Vertical,  and  E  or  W  according  as  it  is  east  or  west  of  the  Meridian. 

b)  By  Tabular  Inspection.—  Divide  the  Alt.  Diff.  by  the  H.  A.  or  W.  T. 
Interval,  as  the  case  may  be,  the  seconds  of  both  being  reduced  to  deci- 
mals of  the  minute,  and  the  result  will  be  the  Change  of  Alt.  in  im. 
With  this,  to  the  nearest  tenth  of  the  minute,  and  the  Lat.  enter  Tab. 
LI,  interpolating  for  the  tenths,  and  the  corresponding  tabular  number 
will  be  the  True  Azimuth,  which  mark  as  prescribed  in  the  preceding 
Rule. 

1  It  is  of  course  necessary  that  the  rate  of  the  watch  should  be  sensibly  uniform  during 
the  Interval. 

3  These  Hour-  Angles  would,  however,  generally  be  deduced  among  the  Data  for  the 
direct  use  of  the  observations  in  question. 


42 


FINDING    THE    COMPASS-ERROR. 


54.  Examples  of  Finding  the  Compass-Error  by  a 
Transition-Azimuth. 


Ex.  1.— At  sea,  in  Lat.  36°  N,  with  an  I 
observation  of  the  Sun  east  of  the  Merid. 
and  south  of  the  P.  V.,  the  Alt.  Diff.  was 
5 7'. 5   and  W.  T.  Interval  2om  24".    The 
Coinpass-Az.  at  the  middle  of  the  Time- 
Interval  being    S  2i°.5  E,   what    is  the 
Compass-Error  ? 
By  Computation : 

H.  A.  Int.    2om  24*          cosec  1.0511 
Alt.  Diff.          57'.5  sin      8.2236 

Latitude         36°.o  sec      0.0920 

True  Az.     S  13  .4  E       sin      9.3667 
Comp.  Az.  S  21  .5  E 
COMP.  ER.        8  .1  E 
By  Inspection : 

Dividing  57'. 5  by  2om.4  we  get  2'. 8, 
as  the  Change  of  Alt.  in  im.  Then,  enter- 
ing Table  LI,  we  find  9°.5  as  the  next  less 
Azimuth,  with  a  Tab.  Diff.  of  4°.8,  of  which 
0.8  part  is  3°.8  (=4°.8  X  0.8) ;  and  hence, 


adding  to  9°.5,  we  get — 

TRUE  Az.  S  i3°.3  E 
Ex.  2.— At  sea,  in  Lat.  51°  35'  N,  an 
observation  of  the  Sun,  W  and  S,  gave  the 
Alt.  Diff.  38'.6  for  a  W.  T.  Int.  of  i2m  43s. 
The  Compass-Az.   being  S  3°-5    E,  what 
is  the  Compass-Error  ? 
By  Computation : 

H.  A.  Int.     I2m438         cosec  1.2561 
Alt.  Diff.          38'.6  sin      8.0511 

Latitude       51°  35'          sec      0.2066 
True  A z.     S  190  .o  W     sin      9.5138 
Comp.  Az.  S  3    .5  E 
COMP.  ER.     22   .5  E 
By  Inspection  : 

Dividing  38'.6  by  12™. 72  we  get  3'.o  to 
the  nearest  tenth  of  the  minute.  Then, 
entering  Table  LI,  we  find— 

TRUE  Az.  S  18°.;  W 


G.  DEPENDENCE  TO  BE  PLACED  ON  THESE  METHODS. 


55.  Possible  Errors  of  the  Azimuth-Data.— The  Data 
employed  in  finding  the  True  Azimuth  are  always  liable  to  be  in  Error, 
more  or  less,  according  to  circumstances.     Consequently,  the  resulting 
Azimuth,  as  obtained  by  either  of  the  preceding  methods,  will  be  cor- 
respondingly uncertain  ;  and,  to  the  end  that'we  may  be  able  to  form  an 
intelligent  opinion  of  the  dependence  to  be  placed  on  the  results,  it  will 
be  necessary  to  consider  the  several  relations  of  the  Data,  as  well  as 
their  possible  Errors,  under  the  variable  conditions  of  experience. 

56.  Estimating    the    Data-Errors;    their    Ordinary 
Limits. — The  Data  are  uncertain  by  amounts  which  are  different,  not 
only  among  themselves  but  at  diiferent  times.    What  these  uncertainties 
are,  on  a  given  occasion,  can  only  be  a  matter  of  special  estimate  under 
the  circumstances  of  the  case.     The  following  considerations  may  serve 
to  fix  our  ideas  on  this  subject : 

a]  The  Latitude-Error. — On  shore  it  is  not  difficult  for  tbe  Navigator 
to  obtain  his  Latitude  within  i',  by  suitable  observations  to  which  he  is 
accustomed. 

At  sea,  when  the  Latitude  is  known  from  a  recent  observation,  it  is 
commonly  reliable  within* 2'  to  5',  at  the  most ;  when  found  through  the 
Reckoning,  after  a  lapse  of  i2h  to  i8h  from  a  preceding  observation,  it 
is  frequently  uncertain  by  5'  to  10',  and  when  obtained,  after  a  lapse  of 
i8h  to  36h  without  an  observation,  the  uncertainty  may  be  not  less  than 
15'  to  30'.  In  cases  of  thick  and  stormy  weather,  no  reliable  estimate 
can  be  formed  as  to  the  Limit  of  the  Latitude- Err  or. 


DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS.  43 


b)  Declination-Error. — With    a  chronometer,  which   will  allow   the 
Greenwich  date  to  be  had  within  30*  or  om.5,  the  Declination,  as  taken 
from  the  Nautical  Almanac,  may  be  depended  on  within — 

i"  for  the  Sun,  the  Planets,  and  Fixed  Stars ; 

7"  to  10"  for  the  Moon. 

On  the  other  hand,  if  the  G.  D.  is  obtained  from  the  Longitude  and 
Ship-Time,  the  Declination  will  be  more  uncertain.  But,  even  then, 
when  the  G.  I),  is  reliable  within  2m,  or  30',  of  Longitude,  the  Declination 
may  still  be  depended  011  within — 

2."  to  3"  for  the  Sun,  the  Planets,  and  Fixed  Stars; 

25"  to  35"  for  the  Moon. 

These  are  Limits  for  the  extreme  cases  of  greatest  change  in  the 
Declination  ;  but,  in  proportion  as  the  G.  D.  is  more  or  less  uncertain 
than  the  Limits  above  stated,  the  Declination  will  be  more  or  less 
uncertain. 

c)  The  IIour: Angle  Error. — Since  the  Hour- Angle  depends  principally 
on  the  Local  or  Ship  Time,  it  follows  that  the  former  cannot  be  expected 
to  be  more  accurate  than  the  latter.    In  reality,  for  the  Sun,  the  Error 
of  tbe  Hour- Angle  is  identical  with  that  of  the  Ship  Apparent  Time,  to 
which  it  corresponds. 

For  the  Moon,  the  Planets,  and  Fixed  Stars,  the  Error  of  the  Hour- 
Angle,  composed  mainly  of  the  Error  in  the  Ship-Time,  is  also  affected 
by  the  small  Errors  of  the  Eight  Ascensions  employed  in  getting  the 
Hour- Angle  (20,  fc).  But,  for  practical  purposes,  the  influence  of  the 
latter,  even  in  the  case  of  the  Moon,  is  unimportant  in  comparison  to 
that  of  the  Ship-Time;  so  that  it  will  be  sufficiently  exact1  to  regard 
the  Error  of  the  Hour-Angle,  for  any  one  of  these  objects,  as  that  of 
the  Ship-Time  to  which  it  corresponds. 

Now  the  Local  Time,  on  shore,  may,  without  much  difficulty,  be  found 
by  the  Navigator  within  4*.  At  sea,  it  may  be  obtained,  according  to 
circumstances,  by  direct  Observation,  within  io8  to  15*;  while,  if  deduced 
through  the  Reckoning,  without  a  very  recent  observation,  it  may  often 
be  at  fault  by  so  much  as  im  or  2m,  or  even  more. 

d)  The  Altitude-Error. — It  is  not  difficult  for  the  Navigator  on  shore 
to  observe  an  Altitude  within  10",  but  it  is  entirely  otherwise  when  he 
is  at  sea.    Then  he  is  quite  fortunate  if,  under  the  most  favorable  cir- 
cumstances, he  can  rely  on  his  Altitude  from  a  single  observation  within 
i/  of  arc.    Ordinarily,  it  is  doubtful  if  Altitudes  can  be  relied  on,  even 
in  the  day-time,  much  within  2' ;  while,  on  a  rough  sea,  or  during  the 
night,  they  may  be  in  Error  from  4'  to  6'. 

e)  Recapitulation  of  the  Limits  of  the  Data-Errors. — In  recapitulation, 
we  may  take  as  ordinary  Limits  of  the  Errors  of  the  Azimuth-Data,  as 
follows :_, 

1  For  example,  with  a  Ship  T.  Error  of  ±  im  and  a  G.  D.  Error  of  ±  im,  the  greatest 
Error  in  the  R.  A.  of  the  San  would  be  ±os.2,  and  the  greatest  Error  in  the  R.  A.  of  the 
Moon  ±3S ;  so  that  the  several  Errors  and  the  probable  total  would  stand — 

V  (6o8)3  -f  (38)2  +  (o".2)3  =  V  3609.04  =  ±  60*.  i 
or  only  os.i  different  from  that  of  the  Ship  T.  Error  alone. 


44  FINDING    THE    COMPASS-ERROR. 


On  shore — 

Lat.  Error,    within  dLi' 

Dec.  Error,    within  ±o'.o5  for  all  celestial  objects 

H.  A.  Error,  within  i  4s 

Alt.  Error,     within  i  0^5 
At  sea — 

Lat.  Error,     within  ±3'  to  ±30' 

Dec.  Error,    within  ±  i'  for  all  objects 

H.  A.  Error,  within  ±  io8  to  i2m 

Alt.  Error,     within  ±2'  to  i6' 

in  which  the  ±  signs  indicate,  what  must  always  be  understood,  the 
uncertainty  as  to  the  Error  being  one  of  excess  (+)  or  deficiency  ( — ). 

57.  Tables  of  Azimuth-Errors ;  Auxiliary  Tables.— 

It  may  be  expedient  to  estimate  the  effect  of  supposed  or  assumed  Errors 
of  the  Data,  sometimes  before,  but  frequently  after, computing  or  other- 
wise finding  a  True  Azimuth.  In  either  case,  this  may  always  be  done 
accurately  and  quite  simply,  with  the  requisite  Data,  by  logarithmic  com- 
putation. But  it  may  also  be  accomplished  without  any  calculation,  and 
with  sufficient  accuracy,  by  mere  inspection,  with  the  use  of  suitable 
Tables.  Such  is  the  object  of  the  several  Tables  of  Azimuth-Errors. 
They  are  specified  in  the  following  list : 

a)  For  Errors  of  Horizon-Azimuths. 

Tab.  XXVIII  for  a  Lat.  Error  of  12'  or  o°.2. 
Tab.  XXIX  for  a  Dec.  Error  of  6'  or  o°.i. 

b)  For  Errors  of  Time-Azimuths. 

Tab.  XXX1Y  for  an  H.  A.  Error  of  ini  of  time  or  15'  of  arc. 
Tab.  XXXV  for  a  Lat.  Error  of  12'  or  o°.2. 
Tab.  XXXVI  for  a  Dec.  Error  of  6'  or  o°.i. 

c)  For  Errors  of  Altitude- Azimuths. 

Tab.  XLIV  for  an  Alt.  Error  of  6'  or  o°.i. 
Tab.  XLV  for  a  Lat.  Error  of  12'  or  o°.2. 
Tab.  XL VI  for  a  Dec.  Error  of  6'  or  o°.i. 

d)  For  Errors  of  Time- Alt.  Azimuths. 

Tab.  XLIX  for  an  H.  A.  Error  of  om.2  or  12". 
Tab.  L  for  an  Alt.  or  Dec.  Error  of  3'  or  o°.o5. 

4 

In  these  Tables  the  arguments  are  not  generally  the  same  as  the  Data 
of  the  Azimuth,  but  these  are  wholly  or  partially  replaced  by  other 
quantities,  such  as  the  Azimuth  itself,  the  Position-Angle  (Jwt),  the 
Altitude  in  a  Time-Azimuth,  and  the  Hour-Angle  in  an  Altitude-Azi- 
muth, etc.  Hence  the  need  of  auxiliary  Tables  to  furnish  these  quanti- 


DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS.  45 


ties  for  the  convenient  use  of  the  Tables  of  Azimuth-Errors.1  Such  are 
the  following : 

Tab.  XXV.-  Position- Angles  for  Horizon- Azimuths. 

Tab.  XXXII.  Position-Angles  for  Time- Azimuths. 

Tab.  XXXIII.  Altitudes  for  Time- Azimuths. 

Tab.  XLIII.  Position -Angles  for  Altitude- Azimuths. 

With  these  Tables,  and  the  aid  of  the  two  Tables  XXXI  and  XLLI, 
of  Direct  and  Limiting  Azimuths,  it  will  be  easy,  whenever  desired, 
whether  in  advance  of  an  observation  or  afterwards,  to  estimate  the  effect 
of  a  supposed  or  assumed  Datum-Error  upon  the  Azimuth.  If  the  Data- 
Errors  in  a  given  case  are  the  same  as  those  for  which  the  Tables  of 
Azimuth-Errors  are  constructed,  the  Azimuth-Errors  may  be  taken  out 
at  sight ;  if  they  are  not  the  same,  it  will  only  be  necessary  to  take 
multiples  or  submultiples  of  the  Tabular  Errors,  according  as  the  assumed 
Data-Errors  are  larger  or  smaller  than  those  of  the  Tables.  Ordinarily? 
the  Azimuth-Errors  are  given  to  the  nearest  tenth  of  the  degree,  which 
is  sufficiently  exact  in  practice. 

The  result  obtained  in  each  special  instance  will  be  a  Partial  Azimuth- 
Error  for  the  particular  Datum-Error  considered. 

«S8.  To  find  Partial  Azimuth-Errors. — In  order  to  illus- 
trate the  use  of  these  Tables  in  finding  Partial  Az.  Errors,  for  certain 
supposed  or  assumed  Errors  of  the  Data,  we  shall  give  in  succession  the 
Eules  for  taking  out  these  Errors  under  each  Azimuth-Method,  following 
Avith  Examples  of  application  to  the  several  Examples  of  Azimuths  given 
under  that  method. 

a)  Partial  Errors  of  Horizon- Azimuths. 

1.  For  the  Lat.  Error:  Enter  Tab.  XXVIII  with  the  Lat.  and  True 
Az.,  each  to  the  nearest  whole  degree,  and  take  out  the  corresponding 
Error. 

2.  For  the  Dec.  Error:  First  enter  Tab.  XXV  with  the  Lat.  and  Dec. 
and  take  out  the  Pos.  Ang.  roughly  to  the  nearest  degree ;  then,  with 
this,  enter  Tab.  XXIX  and  take  out  the  corresponding  Error. 

3.  Remark :  These  Errors  may  obviously  be  taken  out  as  well  before 
as  after  the  observation  is  made. 

As  Examples  of  application,  we  may  take  those  of  Art.  14,  which  we 
shall  place  in  tabular  form,  including  the  Azimuth-Data,  the  Auxiliary 
Data,  the  Assumed  Data-Errors,  and  the  resulting  Partial  Az.  Errors. 

1  The  Tables  of  Errors  in  all  cases  might  be  constructed  with  the  Azimuth-Data  solely 
as  arguments;  but  as  each,  with  the  exception  of  Horizon -Azimuths,  would  require 
three  arguments,  the  Tables  would  need  to  be  considerably  extended,  and  even  then 
be  less  convenient  to  use  than  these. 


46 


FINDING    THE    COMPASS-ERROR. 


Examples  of  the  Partial  Errors  of  Horizon-Azimuths. 

Az.  Data. 

Aux.  Data. 

Sup.  Data-Errors. 

Par.  Az.  Errors. 

No. 

Prob. 

of 

Tot.  A/. 

Ex. 

Lat. 

Dec. 

T.  Az. 

Pos.  Aug. 

Lat. 
Error. 

Dec. 
Error. 

For  Lat. 
Error. 

For  Dec. 
Error. 

Error. 

1 

11°  N 

23°  N 

§7 

o 

79 

o 
±0.2 

o 
±0.1 

o 
O.O 

0 

±0.1 

o 
±0.1 

2 

25  S 

22   N 

114 

63 

O.I 

0.05 

O.O 

O.O 

O.O 

3 

40  N 

21    S 

119 

46 

0.4 

0.2 

±0.2 

±0.2 

±0.3 

4 

328 

23  S 

62 

55 

0.4 

0.2 

O.2 

O.2 

0.3 

5 

51  N 

28  S 

139 

28 

O.2 

O.I 

o-3 

O.2 

0.4 

6 

69  N 

14  N 

48 

16 

O.I 

0.05 

°-3 

0.2 

0.4 

7 

48  N 

ii  S 

106 

4i 

O.2 

O.O5 

O.I 

O.O 

O.I 

b)  Partial  Errors  of"  Time-Azimuths. 

i.  For  the  H.  A.  Error  :  First, 

enter  Tab.  XXXII  with  the  H.  A.,  Lat., 

and  Dec.,  and  take  out  the  Pos. 

Ang. 

;  then,  with  the  II.  A.,  T.  Az.,  and 

Pos.  Aug.,  take  out  the  Az.  Error  from  Tab.  XXXIV. 

2.  For  the  Lat.  Error;  First,  enter  Tab.  XXXIII  with  the  H.  A.,  Dec., 

and  T.  Az.,  and  take  out  the  Alt.  ; 

then,  with  the  T.  Az.  and  Alt.,  take 

out  the  Az.  Error  from  Tab.  XXXV. 

3.  For  the  Dec.  Error  :  .First,  enter 

Tab.  XXXII  with  the  H.  A.,  Lat., 

and  Dec.,  and  take  out  the  Pos. 

Ang. 

;  also,  take  out  the  Alt.  from  Tab. 

XXXIII,  if  not  already  found  (2)5 

then,  ^vith  the  Pos.  Ang.  and  Alt., 

take  out  the  Az.  Error  from  Tab.  XXXVI. 

4.  Remark:  If  the  T.  Az.  be  unknown,  as  in  the  case  of  finding  the 

Az.  Error  in  advance  of  an  observation,  take  it  out  approximately  from 

Tab.  XXXI,  with  the  H.  A.,  Lat, 

and  Dec. 

Examples  of  the  Partial 

Errors  of  Time-  Azimuths. 

Azimuth  Data. 

Auxiliary  Data. 

Partial  Az.  Errors. 

Prob. 

No.  of 

Total 

Ex. 

H.  A. 

Lat. 

Dec. 

Az. 

Pos. 
Ang. 

Alt. 

For  H.  A. 
Err.  ±1™ 

For  Lat. 
Err.  ±12' 

For  Dec. 
Err.  ±tt' 

Az. 

Error. 

h      m 

6 

o 

0 

o 

0 

o 

0 

o 

0 

1 

I      21 

21    N 

3  S 

137 

39 

61 

±0.4 

±0.3 

±0. 

±0.5 

2 

6    35 

41  N 

22    N 

68 

49 

8 

0.2 

O.O 

O. 

O.2 

3 

3    4° 

12    S 

23  S 

70 

91 

33 

O.O 

O.I 

o. 

O. 

4 

5     I0 

6  N 

238 

H5 

65 

8 

±0.1 

0.0 

0. 

O. 

5 

6     10 

23  N 

23  N 

67 

67 

6 

O.I 

O.O 

0. 

O. 

6 

4    3° 

21    N 

21    S 

119 

61 

12 

O.I 

O.O 

o. 

0. 

7 

5      5 

12   N 

12   N 

81 

80 

1S 

0.05 

0.05 

o. 

O. 

* 

9     15 

72  N 

30  N 

36 

12 

14 

O.2 

O.O 

O.O 

0.2 

9 

o     13 

30  N 

9S 

175 

4 

5° 

0.4 

0.0 

0.0 

0.4 

1O 

5    3° 

12   N 

12    S 

103 

76 

5 

O.I 

O.O 

O.I 

O.I 

11 

u    40 

71  N 

21    N 

5 

2 

0-3 

O.O 

0.0 

°-3 

12 

4    36 

i  S 

32  N 

124 

56 

22 

0.2 

•     O.I 

O.I 

°-3 

c)  Partial  Errors  of  Altitude-  Azimuths. 

i.  For  the  Alt.  Error  :  First,  enter  Tab.  XLIII  with  the  Alt.,  Lat.,  and 

Dec.,  and  take  out  the  Pos.  Ang. 

;  then,  with  the  Alt.  and  Pos.  Aug., 

take  out  the  Az.  Error  from  Tab.  XLIV. 

DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS. 


47 


2.  For  the  Lat.  Error :  First,  enter  Tab.  XLYIII  with  the  T.  Az., 
Dec.,  and  Alt.  and  take  out  the  H.  A.;  then,  with  the  Lat.  and  H.  A., 
take  out  the  Az.  Error  from  Tab.  XLV. 

3.  For  the  Dec.  Error:  First,  take  out  the  II.  A.,  if  not  already  taken 
out  (2)  $  then,  with  the  Lat.  and  H.  A.,  take  out  the  Az.  Error  from  Tab. 
XLYI. 

4.  Remark:  If  the  T.  Az.  be  unknown,  as  in  the  case  of  finding  the 
Az.  Error  in  advance  of  an  observation,  take  it  out  approximately  from 
Tab.  XLII,  with  the  Alt.,  Lat.,  and  Dec. 

Examples  of  the  Partial  Errors  of  Altitude-Azimuths. 


Azimuth-Data. 

Auxiliary  Data. 

Partial  Az.  Errors. 

Prob. 

Tolal  Az. 

Ex. 

Alt. 

Lat. 

Dec. 

Az. 

Pos. 
Aug. 

H.  A. 

For  Alt. 
Err.  ±6' 

For  Lat. 
Err.  ±12' 

For  Dec. 
Err.  ±6' 

Error. 

o 

0 

c 

o 

0 

h 

0 

o 

o 

o 

1 

3° 

24  N 

i;N 

84 

70 

4-7 

±0.05 

±0.1 

±0.1 

±0.1 

2 

22 

27S 

10  N 

116 

SS 

3.6 

O.I 

O.2 

0.2 

o-3 

3 

21 

69  N 

14  N 

1  08 

20 

1.4 

0-3 

1.6 

0.9 

1.9 

4 

37 

42  N 

8N 

117 

45 

2.8 

O.2 

°-3 

0.2 

0.4 

d)  Partial  Errors  of  Time-Alt.  Azimuths. 

The  Partial  Errors  are  taken  out  in  a  similar  manner  from  Tables 
XLIX  and  L  for  Time- Alt.  Azimuths,  and  need  not  be  explained  nor 
illustrated  by  Examples. 

59.  Total  Azimuth-Error.— Since  the  signs  of  the  Data- 
Errors  are  uncertain,  it  is  obvious  not  only  that  the  corresponding  Par- 
tial Azimuth -Errors  are  uncertain  in  sign,  but  that  the  Total  Azimuth- 
Error  is  uncertain  in  both  amount  and  sign.  Thus,  the  Partial  Azi- 
muth-Errors may  be  all  +  or  additive,  for  which  the  Total  Azimuth- 
Error  will  be  equal  to  their  Sum  and  also  additive ;  they  may  be  all  - 
or  subtractive,  for  which  the  Total  Azimuth-Error  will  be  equal  to  their 
Sum  and  subtractive;  or  they  may  be  partly  -f  and  partly  — ,  in  which 
case  the  Total  Azimuth -Error  will  be  equal  to  the  Difference  between 
the  Sum  of  the  +  and  the  Sum  of  the  —  Partial  Azimuth-Errors,  and 
will  take  the  sign  of  the  greater  Sum ;  so  that  it  is  even  quite  possible 
for  the  -f-  Errors  to  balance  the  —  Errors  and  leave  no  effective  Total 
Azimuth-Error. 

Now,  when  there  are  several  quantities  of  uncertain  sign,  whose  effect- 
ive total  it  is  desired  to  determine,  we  can  only  resort  to  the  most  prob- 
able estimate.  This  may  be  done,  in  the  present  case,  in  assuming  the 
Errors  to  have  equal  weights,  by  the  following — 

Rule :  The  Probable  Total  of  several  Partial  Errors  is  equal  to  the 
Square  Root  of  the  Sum  of  their  Squares. 

Thus,  Ex.  5  of  Horizon-Azimuths  gives  the  two  Partial  Azimuth- 
Errors  ±o°.3  and  ±o°.2.  Then,  by  the  rule, 

Probable  Total  Az.  Error  = 


V(io°.3)2+ 


or 


48  FINDING    THE    COMPASS-ERROR. 


And  in  this  manner  the  several  Probable  Totals  of  the  last  column  in 
each  of  the  Tables  are  obtained  ;  that  is  to  say,  by  squaring  each  Par- 
tial Azimuth-Error  of  an  Example,  taking  the  Sum  of  those  squares,  and 
extracting  the  Square  Root  of  that  Sum,  taking  the  result  to  the  nearest 
tenth  of  a  degree. 

Of  course,  the  sign  of  the  Probable  Total  is  also  uncertain.  All  that 
we  determine  is  this :  not  knowing  from  the  signs  of  the  Partial  Errors 
whether  we  should  take  a  Sum  or  Difference  of  these  quantities,  we  sim- 
ply get  the  most  probable  mean  opinion  between  extreme  conclusions. 

GO.  Favorable  and  Unfavorable  Conditions.— A  glance 
at  the  Tables  of  Azimuth-Errors  is  sufficient  to  show  how  varied  are  the 
values  of  the  Azimuth-Error  with  respect  to  the  same  Datum-Error. 

Thus,  referring  to  Table  XXXIV  (Errors  of  the  Time-Azimuth  for  an 
Error  of  im  in  the  Hour-Angle),  it  is  seen  that  the  Azimuth-Error  is 
always  comparatively  small  when  the  Azimuth  itself  approaches  to  o°  or 
1 80°,  or  when  the  Position-Angle  is  nearly  90°,  whether  greater  or  less; 
that  the  Azimuth-Error  is  comparatively  large  when  the  Position- 
Angle  approaches  o°  or  180°,  or  when  the  Azimuth  itself  is  nearly 
90°,  whether  greater  or  less.  Also,  that,  with  the  same  Azimuth  and 
Position-Angle,  the  Azimuth-Error  is  small  so  long  as  the  Hour-Angle 
differs  but  little  from  6b;  but  increases,  first  slowly,  at  length  quite 
rapidly,  as  the  Hour- Angle  recedes  from  6h,  becoming  very  large  as  the 
Hour- Angle  approaches  o11  or  i2h. 

These  illustrations  show  that  there  are  certain  relations  between  the 
several  parts  of  the  Triangle  of  Position  (Int.)  which  are  better  than 
others  for  finding  the  Azimuth ;  front  which  it  may  be  said  that  the 
conditions  of  an  Azimuth  are  favorable  or  unfavorable,  according  as  it  is 
affected  to  a  less  or  greater  degree  by  the  same  Errors  of  the  Data.  These 
conditions  of  the  several  Azimuth-Methods  may  be  briefly  stated  in  the 
following  terms: 

a)  Conditions  of  Horizon- Azimuths.— In  these— 

First,  for  an  Error  of  the  Latitude,  the  conditions  are  most  favorable 
when  the  Latitude  is  lowest  and  the  Azimuth  itself  nearest  90°;  they 
are  most  unfavorable  when  the  Latitude  is  highest  and  the  Azimuth 
nearest  o°  or  180°. 

Secondly,  for  an  Error  of  the  Declination,  the  conditions  are  most 
favorable  when  the  Position-Angle  is  nearest  90° ;  they  are  most  unfa- 
vorable when  the  Position- Angle  is  nearest  o°  or  180°. 

Otherwise,  it  may  be  said  that  the  conditions  are  most  favorable  when 
the  Latitudes  are  lowest  and  the  Declinations  least;  they  are  most 
unfavorable  when  the  Latitudes  are  highest  and-  the  Declinations 
greatest. 

b)  Conditions  of  Time-Azimuths.— In  these— 

First,  for  an  Error  of  the  Hour-Angle,  the  conditions  are  most  favor- 
able when  the  Hour-Angle  and  Position-Angle  are  both  nearest  90°,  and 
the  Azimuth  itself  nearest  o°  or  i8o°j  they  are  most  unfavorable  when 


DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS.  49 


the  Hour- Angle  and  Position- Angle  are  both  nearest  o°  or  180°,  and 
the  Azimuth  nearest  90°. 

Secondly,  for  an  Error  of  the  Latitudes,  the  conditions  are  most  favor- 
able when  the  Altitude  of  the  object  is  least  and  the  Azimuth  itself 
nearest  o°  or  180°;  they  are  most  unfavorable  when  the  Altitude  is 
greatest  and  the  Azimuth  nearest  90°. 

Thirdly ',  for  an  Error  of  the  Declination,  the  conditions  are  most  favor- 
able when  the  Altitude  is  least  and  the  Position-Angle  nearest  o°  or 
1 80°  ;  they  are  most  unfavorable  when  the  Altitude  is  greatest  and  the 
Position- Angle  nearest  90°. 

c)  Conditions  of  Altitude-Azimuths.— In  these— 
First,  for  an  Error  of  the  Altitude,  the  conditions  are  most  favorable 
when  the  Altitude  is  least  and.  the  Posi tion- Angle  nearest  90° ;  they 
are  most  unfavorable  when  the  Altitude  is  greatest  and  the  Position- 
Angle  nearest  o°  or  180°. 

Secondly,  for  an  Error  of  either  the  Latitude  or  Declination,  the  con- 
ditions are  most  favorable  when  the  Latitude  is  lowest  and  the  Hour- 
Angle  nearest  6h;  they  are  most  unfavorable  when  the  Latitude  is 
highest  and  the  Hour- Angle  nearest  oh  or  i2h. 

<7)  Conditions  of"  Time-Alt.  Azimuths. — These  conditions 
need  not  be  particularized,  as  they  are  comprised  in  those  stated  under 
the  two  preceding  heads,  and  may  be  inferred  from  a  glance  at  Tables 
XLIX  and  L. 

61.  Limits  of  Allowable  Azimuth-Error. — It  is  com- 
monly regarded  as  "sufficient  for  the  requirements  of  Navigation"  if 
the  Helmsman  is  able  to  steer  within  a  quarter  of  the  point,  or  2°.8  of  the 
course  set  for  him.  Let  this  be  conceded  as  a  limit  for  steering ; 
then,  it  is  also  regarded  by  many  Navigators  as  "  sufficiently  exact,"  in 
view  of  the  Steersman's  margin,  to  take  out  their  True  Azimuths,  when- 
ever it  can  be  done  by  Tables,  to  the  nearest  degree  ;  while  it  is  also  fre- 
quently admitted  that  little  pretension  is  made,  in  the  observation  of 
their  Compass- Azimuths,  to  get  them  more  closely  than  to  the  nearest 
degree. 

Now,  we  have  here  three  distinct  sources  of  Error,  each  of  quite  an 
appreciable  amount,  which  are  allowed  to  occur  on  every  occasion  of 
shaping  and  steering  a  ship's  course,  without  mentioning  the  Error  from 
Defective  Sensibility,  which,  although  nothing  for  some  compasses,  may 
be  several  degrees  for  others.  There  are,  then,  in  supposing  the  com- 
pass to  be  practically  perfect,  as  follows : 

Pointing-Error  io°.5  in  observing  the  Comp.  Az. ; 

Azimuth-Error  ±o°.5  in  taking  out  from  Tables ;' 

Steering- Error   =t  i°.4  in  steering  the  ship's  course ; 
giving  a  Probable  Total  Error  of  i  i°.6.     It  is  true  that  these  Errors 
might  have  opposite  signs  in  some  cases,  when  the  total  would  be 
reduced  to  o°,4;    but  they  might  with  equal  probability  have  the  same 
signs,  when  the  total  would  be  increased  to  2°.4. 


50  FINDING   THE    COMPASS-ERROR. 


But  it  must  always  be  remembered  that,  as  already  exemplified  in  the 
preceding  Articles,  in  consequence  of  the  inevitable  Errors  of  the  Data, 
there  is  quite  likely  to  be  an  Error  of  the  True  Azimuth,  however  care- 
fully it  may  be  computed  or  taken  out  from  the  Tables.  This  may  be 
quite  small  under  very  favorable  conditions  j  but  it  may  also  be  quite 
large  under  different  circumstances. 

Hence,  on  the  simple  principle  that,  if  "  we  cannot  save  everything, 
we  should  strive  to  save  all  we  can,"  it  would  seem  to  be  obvious  that 
it  is  always  advisable  to  avoid  all  unnecessary  Errors,  or  to  diminish 
them,  at  the  least,  whenever  we  may. 

Without  in  this  place  entering  further  on  the  question  of  the  need 
to  allow  so  large  a  margin  for  Steering-Error,  or  so  large  an  Error  of 
Observation,  it  may  be  remarked,  with  respect  to  the  two  sources  of  the 
True  Azimuth  Error,  that  it  has  already  been  shown  to  be  easy  enough, 
by  the  methods  described,  to  take  out  the  True  Azimuth  from  the 
Tables,  within  ±o°.2  to  io°.3j  while,  of  the  Examples  given  of  Az. 
Errors  due  to  the  Errors  of  the  Data  (58,  59),  it  is  seldom  that  the  Prob. 
Tot.  Azimuth-Error  from  this  source,  with  the  actual  conditions,  need 
exceed  =L  o°.4.  It  is  evident,  therefore,  that  we  should  be  able,  in  find- 
ing any  Single  Compass-Error,  to  depend  on  our  True  Azimuth,  within — 

V(o°.3)2+(o0.4)2  =  ±o°.5 
for  all  ordinary  cases  of  requiring  it  at  sea. 

On  the  other  hand,  in  the  operation  of  finding  Serial  Compass-Errors 
for  Different  Headings  of  the  Ship,  it  is  necessary  and  sufficient  that  for 
port  or  inshore  observations  our  True  Azimuth  should  be  depended  on 
within  one-tenth  of  a  degree,  or  dLo°.i ;  while  for  observations  at  sea  we 
must  be  content  to  get  the  best  we  can ;  in  general,  however,  we  may 
expect  to  get  these  within  three-tenths  of  the  degree,  or  i  o°.3. 

Accordingly,  we  shall  take  these  as  Requirement- Limits  in  those 
cases  of  nautical  practice  within  which  True  Azimuths  are  to  be 
employed  in  finding  the  Compass-Error  $  that  is  to  say : 

1.  For  the  Single  Compass-Error,  the  greatest  Error  of  Azimuth  not 
to  exceed  =t  o°.5  ; 

2.  For  Serial  Compass-Errors,  the  greatest  Error  of  Azimuth  not  to 
exceed  rto°.i  in  port,  or  ±o°.3  at  sea. 

In  order  to  satisfy  these  requirements  in  practice,  we  must  be  assured, 
in  the  first  place,  that  our  Data-Errors  (56)  in  any  given  case  are  cer- 
tainly within  the  limits  assumed  for  them,  whatever  those  may  be ;  and, 
secondly,  that  the  conditions  of  the  Azimuth  give  rise  to  Partial  Azi- 
muth-Errors, as  found  by  a  reference  to  the  proper  Tables  (57),  whose 
Probable  Total  is  within  the  assigned  Eequirement-Limit. 

By  these  means  we  shall  be  able  to  make  a  reliable  estimate  of  the 
dependence  to  be  placed  on  our  Compass-Observations,  so  far  at  least 
as  finding  the  True  Azimuth  is  concerned,  as  well  in  advance  as  after- 
wards ;  so  that,  if  the  results  cannot  be  depended  on  within  the  required 
limits,  we  may  secure  more  favorable  conditions,  either  by  choosing  a 
different  object,  or  waiting  for  a  favorable  change ;  or  so  that,  at  the 
least,  we  may  take  our  succeeding  steps  with  intelligence  and  certainty. 


DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS.  51 


We  shall  next  proceed,  in  further  elucidation  of  this  important  sub- 
ject, with  the  aid  of  a  series  of  Tables  giving  the  Limiting-Errors  of 
Azimuths  for  certain  assumed  limits  of  the  Data-Errors,  to  show  how  we 
may  safely  estimate,  in  any  given  case,  the  dependence  to  be  placed 
upon  either  of  the  several  preceding  methods  of  finding  the  True  Azi- 
muth. 

62.  Dependence  on  Horizon-Azimuths :  Use  of*  Tab. 

XXVI. — Since  an  Horizon- Azimuth  can  only  be  resorted  to  for  a  Sin- 
gle Compass-Error,  it  is  unnecessary  to  consider  the  second  Bequirement- 
Limit.  If,  then,  these  Azimuths  be  restricted  to  the  Sun,  Moon,  Planets, 
and  those  Stars  whose  Declinations  never  exceed  about  30°  north  or 
south;  and  if  we  assume  a  Lat.  Error  within  ±12',  or  io  .2,  and  a 
Dec.  Error  within  =L6',  or  ±o°.i,  Tab.  XXVI  may  be  formed  directly 
from  Tables  XXIV,  XXV,  XXVIII,  and  XXIX. 

This  Table  consists  of  four  parts,  corresponding  to  the  Declinations  of 
12°,  1 8°,  24°,  and  30°;  the  first  and  second  columns  of  each  division  con- 
taining the  greatest  Par.  Az.  Errors,  to  the  nearest  superior  tenth  of  the 
degree,  for  the  assumed  Lat.  and  Dec.  Errors,  answering  to  the  Lati- 
tudes in  the  left-hand  column,  to  which  they  are  set  opposite.  The  third 
column  in  each  division  contains  the  Probable  Total  Az.  Errors ;  each 
of  which  being  found  by  taking  tne  Square  Boot  of  the  Sum  of  the 
Squares  of  the  corresponding  Par.  Az.  Errors. 

It  is  thus  seen  that,  with  the  Data-Errors  within  the  limits  here 
stated,  and  a  Dec.  not  greater  than  12°,  the  Prob.  Tot.  Error  of  an 
Horizon -Azimuth  will  not  exceed  ±o°.2  for  all  Latitudes  up  to  60° 
N  or  S;  not  exceed  ±o0.6  for  all  Latitudes  between  60°  and  70°; 
nor  exceed  ±i°.o  even  up  to  76°  of  Latitude.  But,  as  the  Dec.  is 
greater,  the  corresponding  Az.  Errors  range  at  higher  limits;  so  that, 
for  example,  with  a  Dec.  of  30°,  the  Az.  Error  might  be  ±o0.7  for  a  Lat. 

of5S°- 
The  small  Table  at  the  right  is  a  similar  one  for  all  Declinations  up 

to  24°,  in  which  the  Lat.  and  Dec.  Errors  are  each  assumed  to  be  ±o°.5, 
or  ±30'.  It  shows  the  Prob.  Tot.  Error  to  which  one  is  liable,  when 
careless  or  indifferent  about  the  Data,  either  in  their  preparation,  or  in 
taking  out  from  the  Azimuth-Tables. 

It  is  not  an  uncommon  practice  with  Navigators  to  enter  the  Table  of 
Horizon- Azimuths  (or  Amplitudes)  with  the  Lat.  and  Dec.  to  the  nearest 
whole  degree,  on  the  ground  of  greater  ease  in  taking  out  the  required 
quantity.  If  these  Data  could  be  supposed  precisely  correct,  then,  in 
using  them  to  the  nearest  degree,  we  should  involve  at  the  most  an  Error 
of  ±o°.5  in  each;  and,  accordingly,  the  small  Table,  before  referred  to, 
would  give  the  corresponding  greatest  Prob.  Tot.  Errors  to  which  such 
a  practice  would  be  liable.  Thus,  we  might  still  depend  on  our  True 
Azimuths  within  ±  i°.o  for  all  Latitudes  up  to  50°,  BT  or  S. 

If,  on  the  other  hand,  as  is  most  certain  to  be  the  case,  the  Data  are 
both  in  Error,  the  Lat.  especially  being  sometimes  uncertain  by  12',  20', 
or  even  30',  we  may  involve  the  Lat.  in  Error,  by  at  least  a  whole  degree, 
in  taking  it  to  the  nearest  degree.  Hence,  the  Par.  Az.  Errors  in  the 


52  FINDING    THE    COMPASS-EKKOK. 


second  column  of  the  small  Table  might  be  doubled,  and  the  Prob.  Tot. 
Error  of  the  True  Az.  for  a  Lat.  of  50°  might  be  ±  i°-4. 

63.  Dependence  on  Time-Azimuth* :  Use  of  the 
Tables  of  Limiting-Errors. — Regarding  Time-Azimuths  as 
available,  alike  in  finding  Single  and  Serial  Compass-Errors,  we  shall 
consider  the  True  Azimuths  with  respect  to  both  Requirement-Limits. 

a)  Single  Time-Azimuth*  within  the  First  Require- 
ment-Limit.— In  order  to  furnish  some  practical  illustrations  of  the 
Errors  to  which  a  Time- Azimuth  may  be  liable,  under  different  possible 
circumstances  at  sea,  and  thus  establish  certain  Limits  of  Dependence, 
three  distinct  suppositions  of  Data-Errors  will  be  made,  as  follows  : 

1.  Ordinary  or  fair-weather  sailing,  but  with  some  dependence  on  the 
D.  R; 

2.  Cloudy,  without  regular  or  reliable  observations  for  Time  and 
Position,  with  increased  dependence  on  the  Reckoning ; 

3.  Thick  weather,  but  ship  kept  nearly  to  her  course. 

Evidently  any  assumptions  of  Data-Errors,  under  these  supposed  con- 
ditions, must  be  mainly  arbitrary;  but  this  will  be  of  little  consequence, 
with  the  use  to  be  made  of  these  TaJbles. 

Tables  XXXVII  and  XXXVIII  are  formed  in  accordance  with  these 
suppositions.  The  former  answers  to  all  Latitudes  from  o°  to  60°  N  or  S, 
and  for  all  Declinations  from  o°  to  30°  of  the  same  or  contrary  name; 
while  the  latter  answers  to  all  Latitudes  between  60°  and  80°,  and  for  the 
same  range  of  Declinations.  The  greatest  Par.  Az.  Errors  are  given  in 
each  division  of  the  Tables  for  the  H.  A.  Limits  set  down  in  the  first 
column. 

From  this  Table  the  following  conclusions  may  be  drawn  : 

First.  With  the  Data-Errors  limited  according  to  the  first  supposition, 
a  Time-Azimuth  of  any  object  may  be  had  during  the  day  or  night, 
without  being  liable  to  a  greater  Probable  Error  than  ±  i°.o,  whatever 
the  Lat.  up  to  80°,  even  when  the  least  H.  A.  is  no  more  than  ih.  As  the 
limiting  H.  A.  is  greater,  the  Azimuth-Error  becomes  less,  until  it 
comes  to  be  less  than  ±  o°.3,  for  all  H.  Angles  greater  than  7h.  Moreover, 
for  all  H.  Angles  in  Polar  Latitudes,  and  for  all  H.  Angles  greater  than 
3h  in  all  other  Latitudes,  the  True  Azimuth  may  be  depended  on  within 
±o'.S. 

Secondly.  With  the  Data-Errors  limited  by  the  second  supposition,  the 
Prob.  Tot.  Azimuth-Error  does  not  exceed  io°.7,  for  all  Latitudes,  pro- 
vided no  H.  A.  smaller  than  3l1  is  admitted.  And  in  this  case  the 
greatest  Prob.  Azimuth-Error  is  less  than  io°.5  for  all  H.  Angles 
greater  than  8h. 

Thirdly.  Even  with  the  Data-Errors  so  large  as  the  limits  of  the  third 
supposition,' an  uncertainty  of  the  Data  which  should  never  exist  except 
under  severe  stress  of  weather,  the  True  Azimuth  may  still  be  depended 
on  within  ±  o°.S,  provided  the  H.  A.  of  the  object  is  not  allowed  to  be  less 
than  51'. 


DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS.  53 


In  reality,  however,  all  the  extreme  conditions  of  Tables  XXXVII 
and  XXXVIII  very  seldom,  if  ever,  concur  at  the  same  time ;  on  the 
contrary,  it  most  commonly  happens  that  the  conditions  are  much  more 
favorable,  even  with  the  same  Errors  of  Data.  Consequently,  the  Azi- 
muth-Errors for  the  same  Data-Errors  will  generally  be  smaller  than  the 
maximum  or  limiting  values  of  this  Table. 

This  is  seen  in  the  Examples  of  Time- Azimuths  (23,  25),  which  repre- 
sent a  considerable  variety  of  actual  conditions,  and  whose  Errors  are 
given  in  the  Table  of  Art.  58,  for  assumed  Data-Errors  in  accordance 
with  the  first  supposition  of  Table  XXXVII. 

Table  XXXVIII  illustrates  the  convenience  with  which  Time-Azi- 
muths are  adapted  to  the  wants  of  the  Navigator,  even  in  the  highest 
Latitudes.  Thus,  it  is  seen  that  the  probable  total  effect  upon  the  Azi- 
muth of  quite  large  Data-Errors  is  not  only  within  the  limits  of  ±o°.5, 
but  is  nearly  uniform  for  all  Hour- Angles  in  the  entire  circle,  from  oh  to 
i2h,  E  or  W  of  the  Meridian. 

b)  Serial  Azimuths  within  the  Second  Requirement- 
Limit. — Unlike  the  ordinary  case,  in  which  the  Navigator  may  desire 
to  find  his  Compass-Error  with  every  important  change  of  course,  and 
where  he  is  compelled  to  accept  the  situation  and  the  best  available 
result,  whatever  the  circumstances,  in  the  case  of  getting  a  set  of  Serial 
Compass-Errors,  he  may  generally  choose  his  opportunity  and  thus  com- 
mand the  best  obtainable  conditions. 

When  in  port,  or  otherwise  near  the  land,  the  ship's  Position  and 
Time  are  always  known  or  may  be  obtained  with  considerable  precision. 
Thus,  the  Navigator  may  generally  find  his  Time  and  Latitude,  if  not 
otherwise  available,  the  former  within  ±48  and  the  latter  within  ±  i',  by 
his  own  observations  on  shore. 

At  sea,  moreover,  in  all  Latitudes  from  the  Equator  to  60°  north  or 
south,  with  a  proper  choice  of  circumstances,  suitable  for  Serial  Azi- 
muths, the  Navigator  should  be  able  to  get  his  Time  within  i  ios  to  ±  is8 
and  his  Latitude  within  dL  3',  by  excluding  all  dependence  on  the  Reckon- 
ing. Beyond  the  Latitude  of  60°,  the  Time  will  be  found  with  less  pre- 
cision, but  the  Latitude  with  somewhat  greater  certainty. 

Table  XXXIX  is  formed  with  reference  to  these  supposed  circum- 
stances. The  Declination  is  taken  at  a  limit  of  25°  N  or  S,  for  the  Sun 
and  other  zodiacal  bodies ;  and  the  greatest  Prob.  Azimuth-Errors  are 
given  in  each  division  of  the  Table  for  the  H.  A.  Limit,  set  down  in  the 
first  column. 

From  this  Table,  in  assuming  the  Data-Errors  to  be  kept  within  the 
limits  here  given,  the  following  conclusions  may  be  drawn : 

First,  that,  while  in  port  or  near  the  land,  in  all  Latitudes,  Serial 
Time- Azimuths  may  be  taken  with  any  Hour- Angle,  not  less  than  2^h 
from  the  Meridian,  without  being  liable  to  a  greater  Probable  Error  than 
±3',  or  io°.o5. 

Secondly,  that,  anywhere  at  sea,  within  the  parallels  of  80°  N  and 
80°  S,  Serial  Time-Azimuths  may  be  taken  within  the  same  Limit  of  the 


54  FINDING   THE    COMPASS-ERROR. 


Hoar- Angle,  under  ordinary  circumstances,  without  being  liable  to  a 
greater  Probable  Error  than  ±7',  or  io°.i2. 

Thirdly,  that,  anywhere  at  sea,  Serial  Time- Azimuths  may  be  had 
within  the  same  Limit  of  Hour-Angle,  without  being  liable  to  a  greater 
Probable  Error  than  ±  16',  or  rt  o°.3,  even  when  the  Data  are  considerably 
uncertain,  as,  for  example,  with  the  Data-Errors  assumed  in  the  third 
division. 

Keeping  in  mind,  however,  that  the  Azimuth-Errors  set  down  in  these 
Tables  correspond  to  extreme  conditions,  it  will  be  seen  that  it  is 
quite  within  the  reach  of  the  Navigator's  appliances  and  of  the  condi- 
tions under  his  control  to  obtain  a  series  of  True  Azimuths,  by  this 
method,  sufficient  for  a  complete  circle  of  different  Headings  of  the  ship, 
in  whatever  part  of  the  sea  likely  to  be  traversed  by  him,  which  may  in 
general  be  depended  on  within  two  to  three  tenths  of  the  degree. 

Nor  need  the  Navigator  confine  his  observations  to  the  Sun.  For 
the  Planets  and  Fixed  Stars  may  be  observed  with  equal  accuracy,  so 
long  as  the  Time  and  Latitude  are  known  within  the  above-mentioned 
Limits,  and  the  observations  are  made  under  similar  restrictions  as  to 
the  smallness  of  the  Hour- Angle.1 

The  Moon,  with  the  exception  of  a  little  additional  labor  in  prepar- 
ing her  Hour- Angles  (20,  21),  may  be  used  with  a  facility  hardly 
inferior  to  that  in  using  the  Sun,  if  taken  during  the  period  of  greatest 
Declination.  And,  at  other  times,  even  when  her  Dec.  changes  most 
rapidly,  a  short  or  partial  series  of  Azimuths  may  be  had  from  this 
object,  when,  for  a  period  of  half  an  hour,  the  Error  of  the  Middle  Dec. 
need  not  exceed  4^';  a  condition,  that  should  still  give  us  results  which 
may  be  depended  on  within  ±o°.3  or  ±o°.4,  quite  sufficient  for  an  imme- 
diate stress  of  circumstances  which  should  compel  a  resort  to  this  object 
in  the  absence  of  one  more  favorably  conditioned. 

64.  Dependence   on   Cireumpolar   Azimuths. — Table 
XL,  while  giving  direct  values  of  the  True  Azimuth  of  Polaris,  also 
furnishes  at  sight  the  requisite  indications  of  the  Errors  due  to  un- 
certainties of  the  Data.    And,  with  respect  to  these  Errors,  this  Table 
illustrates,  in  a  striking  manner,  how  nearly  complete  in  its  independ- 
ence, is  this  method  of  getting  a  True  Azimuth,  of  all  its  Data.     Not  only 
is  it  sensibly  free  from  any  dependence  on  the  Dec.,  but  it  is  practically 
independent  of  the  Lat. ;  while,  as  to  the  Hour-Angle,  represented  by 
the  Sid.  Time,  the  Azimuth  varies  less  than  o°.i  during  an  hour  of  Time, 
when  the  Star  is  near  one  of  its  greatest  Elongations.    And,  during 
other  periods  of  the  apparent  daily  revolution  of  this  Star,  its  changes 
in  Azimuth  are  still  quite  small  in  comparison  to  the  changes  in  its  H. 
A.,  or  in  the  Time  derived  therefrom. 

65.  Dependence  on  Altitude-Azimuths:  Use  of  the 
Tallies  of  Limiting-Errors. — As  already  evident  in  what  has 

1  The  principal  practical  objection  to  the  use  of  these  objects  will  bo  found  in  the 
difficulty  of  sighting  them  with  sufficient  distinctness  when  near  the  Horizon,  and  the 
other  attending  inconveniences,  especially  of  night-observations. 


DEPENDENCE    TO    BE    PLACED    ON    THESE    METHODS. 


55 


preceded,  but  will  be  further  referred  to  in  a  succeeding  Article, 
the  computations,  as  well  as  the  observations  of  Altitude- Azimuths,  are 
less  convenient  than  those  of  Time-Azimuths.  Especially  is  this  true 
of  Serial  Azimuths  by  this  method.  We  shall  also  see,  from  the  follow- 
ing considerations,  that  their  Errors  develop  more  rapidly  under  certain 
conditions  than  those  of  the  latter. 

a)  Single  Azimuths  within  the  First  Requirement- 
Limit* — In  Table  XLYII  are  given  the  greatest  Azimuth-Errors  rela- 
tive to  certain  assumed  Data-Errors,  for  a  series  of  different  Latitudes, 
under  three  different  limitations  of  the  smallest  Hour- Angle,  but  answer- 
ing also  to  the  corresponding  Supplements,  or  largest  Hour-Angles. 
They  are  taken  out  for  a  Declination  of  30°,  and  therefore  cover  all  Dec- 
linations of  the  zodiacal  bodies,  like  the  Sun,  Moon,  etc. 

Thus,  the  Data-Errors  being  within  the  limits  of  this  Table  so  long  as 
the  Hour- Angle  is  not  less  than  ih  nor  greater  than  nh,  an  Altitude- 
Azimuth  of  any  object  whose  Declination  does  not  exceed  30°,  in  what- 
ever Latitude  from  the  Equator  to  40°,  may.  be  had  with  a  certainty  of 
no  greater  Probable  Error  than  i  i°.2.  Again,  so  long  as  the  Hour- 
Angle  is  not  less  than  2h  nor  greater  than  ioh,  an  Altitude- Azimuth 
may  be  had  in  any  Latitude  up  to  70°  with  no  greater  Probable  Error 
than  i  i°.2,  and  so  on ;  as  the  H.  A.  Limit  is  more  remote  from  the  Me- 
ridian, upper  and  lower,  the  conditions  are  improved,  and  the  maximum 
Probable  Error  diminished. 

/>)  Serial  Azimuths  within  the  Second  Requirement- 

Limit.—  Altitude- Azimuths  are  occasionally  resorted  to  in  serial  obser- 
vations for  Compass-Error  5  and  it  is  accordingly  advisable  to  consider 
the  question  of  their  liability  to  Error,  and  how  far  they  can  be  depended 
on,  when  employed  for  this  purpose. 

The  subjoined  little  Table  gives  the  Errors  of  Altitude- Azimuths  under 
the  same  conditions  as  those  of  Table  XLVII,  except  that  the  several 
Data-Errors  are  reduced  to  their  minimum  limits  for  favorable  circum- 
stances at  sea,  and  the  Hour- Angle  Limits  of  approach  to  the  Meridian 
are  made  larger. 

Limiting  Errors  of  Serial  Altitude-Azimuths. 


Par.  Azimuth-Error. 

For  Limits  of 

For  Latitudes 

Prob. 
Total 

Hour  -Angles. 

N  or  S. 

Alt. 

Lat. 

Dec. 

Az. 

Error 

Error 

Error 

Error. 

±1' 

±3' 

±1' 

h        h 

o            o 

( 

/ 

3  to  9 

o  to  30 

±2 

±5 

±2 

±6 

4        8 

30           60 

2 

4 

3 

6 

5        7 

60           70 

3 

3 

3 

6 

5        7 

70           30 

4 

4 

6 

10 

Thus,  if  the  Data-Errors  may  be  safely  assumed  within  i  i'  for  the 
Altitude,  ±3'  for  the  Latitude,  ±  i'  for  the  Declination,  the  correspond- 
ing Partial  Azimuth-Errors  will  not  exceed  those  set  down,  and  the 


56  FINDING   THE    COMPASS-ERROR. 


Prob.  Tot.  Error  of  the  Azimuth  will  not  be  greater  than  ±  6'  for  all 
Latitudes  from  o°  to  70°,  provided  the  Hour- Angles  are  kept  within  the 
limits  stated  in  the  first  column  of  the  Table.  For  Latitudes  beyond  70°, 
the  Total  Errors  will  exceed  ±6',  for  an  Hour- Angle/  even  if  pre- 
cisely 6h. 

66.  Relative  Advantages  of  these  Methods. — We  shall 
now  consider  very  briefly  the  relative  advantages  of  these  methods. 

a)  Horizon- Azimuths. — This  method,  in  the  well-known  form  of  Ampli- 
tudes, has  long  been  used  by  Navigators,  generally,  as  a  means  of  find- 
ing the  Compass-Error.  The  facility  with  which  the  observation  is 
made,  and  the  convenience  with  which  the  Table  is  used,  have  alike 
contributed  to  make  this  a  popular  method. 

Whether  in  the  form  of  Azimuths,  as  here  given,  or  in  that  of 
Amplitudes,  as  heretofore  used,  it  has  been  shown  to  admit  of  suffi- 
ciently precise  results  under  a  great  variety  of  circumstances  (62);  and, 
with  the  provision  of  observing  the  object  in  the  Apparent  Horizon  (7),  it 
is  equally  available  in  observations  of  the  Moon,  and  thereby  its  practi- 
cal utilities  are  largely  extended. 

This  method  is  subject,  however,  to  the  following  disadvantages : 

1.  It  is  unavailable,  except  at  certain  moments  of  the  day  or  night.2 

2.  It  is  entirely  unavailable  in  high  Latitudes,  with  respect  to  bodies 
which  remain  above  the  Horizon,  or  do  not  set. 

3.  Even  when  available  for  certain  objects  in  high  Latitudes,  it  is  lia- 
ble to  large  Errors,  and  must  be  used  with  caution. 

1))  Time- Azimuths. — Time- Azimuths  have  the  following  advantages: 

1.  In  the  simplicity  and  ease  of  the  observation,  which  even  exceeds 
that  of  the  Horizon-Azimuth,  or  Amplitude. 

2.  In  the  facility  and  precision  with  which  it  may  be  employed  in 
serial  observations  of  Compass-Error,  whether  in  port  or  at  sea. 

3.  In  often  being  available  when  neither  of  the  other  methods  can  be 
employed,  as  in  the  case  of  an  obscured  or  ill-defined  Horizon. 

4.  In  being  available  whenever  any  celestial  body  is  distinctly  visible 
at  a  convenient  Altitude  for  a  compass-bearing,  whether  Sun,  Moon, 
Planet,  or  Fixed  Star. 

5.  In  being  available  even  when  the  Time  and  Latitude  are  uncertain 
to  a  very  considerable  extent,  by  a  suitable  choice  of  object,  in  order  to 
avoid  unfavorable  conditions. 

A  prejudice  against  this  method  seems  to  have  existed  to  a  consider- 
able extent,  even  among  intelligent  Navigators.  Doubtless  this  has 
been  due  in  some  degree  to  the  supposed  greater  labor  of  computation 
and  perplexity  of  cases.  Nevertheless,  the  computation,  even  of  a 

^he  consideration  of  the  H.  A.  Limit  in  Alt.  Az.  Errors  has  been  adhered  to  in  the 
text,  not  only  because  it  conveniently  enters  into  the  estimate  of  these  Errors,  but  for 
the  purpose  of  a  comparison  with  the  conditions  of  Time-Azimuths. 

2This  objection  may  not  be  very  serious  with  those  Navigators  who  depend  soldy  on 
morning  and  evening  Amplitudes  of  the  Sun  for  their  Compass-Errors. 


DEPENDENCE    TO    BE    PLACED    ON   THESE    METHODS.  57 


Time- Azimuth,  is  really  quite  as  short  and  convenient  as  that  of  an 
Altitude- Azimuth  of  the  same  body,  more  commonly  employed. 

But  the  simplicity  and  certainty  of  this  method  are  still  further  assured 
with  the  use  of  the  Time-Azimuth  Tables.  By  these  means  the  operations 
of  taking  out  an  Azimuth  may  commonly  be  done  at  sight,  with  suffi- 
cient accuracy,  in  ordinary  cases  of  finding  the  Compass-Error ;  while  it 
is  always  easy  to  find  the  Azimuth  to  the  nearest  tenth  of  the  degree, 
whenever  required  for  a  series  of  different  Headings  of  a  ship. 

c)  Circumpolar  Azimuths. — This  modification  of   the  Time- Azimuth 
Method,  subject  only  to  the  condition  that  the  Star  (Polaris)  can   be 
conveniently  observed   through  the  sight- vanes  of  the  azimuth-circle, 
must  be  especially  useful  whenever  the  Hour- Angle  and  Latitude  are 
very  uncertain,  as  they  not  un frequently  are  after  long-continued  thick 
weather,  with  entire  absence  of  observations,  and  the  Reckoning  even 
very  much  at  fault. 

d)  Altitude- Azimuths. — Altitude- Azimuths  have  the  following  disad- 
vantages: 

1.  In  the  necessity  of  measuring  the  Altitude  of  the  object,  in  addi- 
tion to  that  of  observing  the  Compass-Azimuth. 

2.  In  being  inconvenient  for  serial  observations  of  Compass-Errors, 
from  the  much  greater  labor  of  the  requisite  observations  and  computa- 
tions. 

3.  In  being  frequently  unavailable,  from  the  difficulty  of  observing  an 
Altitude,  in  consequence  of  an  obscured  or  ill-defined  Horizon. 

4.  In  being  liable  to  rapidly-increasing  Error  in  the  higher  Latitudes 
from  the  Errors  of  the  Data. 

In  view  of  these  disadvantages,  without  any  very  obvious  compensa- 
tion, there  would  see  in  to  be  little  occasion  for  resorting  to  the  observa- 
tion of  an  Altitude,  merely  to  find  the  Compass-Error ;  since,  in  general, 
whenever  an  Altitude- Azimuth  is  available,  a  Time-Azimuth  would  be 
both  available  and  preferable. 

If,  however,  the  Navigator  would  follow  the  commendable  practice 
of  taking  the  Compass-Bearing  of  the  object,  whenever  practicable,  in 
every  observation  made  for  Time  and  Position,  then  the  Altitude  ob- 
tained for  other  purposes  might  be  used  in  an  Altitude- Azimuth,  for 
finding  the  Compass-Error  on  the  same  occasion.  Even  in  this  case  the 
Time-Azimuth  might  still  be  preferred,  except  with  the  somewhat  rare 
occurrence  that  the  Navigator  has  a  very  uncertain  Ship-Time  but  a 
good  Latitude,  in  which  case  the  Altitude-Azimuth  might  give  a  more 
reliable  result. 


58  FINDING    THE    COMPASS-ERROR. 


II. — OBSERVATIONS  OF  TERRESTRIAL  OBJECTS. 
A.   METHOD  BY  DIRECT  BEARINGS. 

The  Method  by  Direct  Bearings  consists  in  observing 
the  Compass-Bearing  of  some  prominent  and  well-defined  object  on  the 
land,  whose  True  Bearing  is  already  known,  or  may  be  found,  from  the 
station  at  which  the  observations  are  made. 

The  difference  between  the  Compass-Bearing  and  True  Bearing  of  the 
object  is  the  Compass  Error  (5). 

67.  Fundamental  Conditions.— There  are  three  requirements 
in  the  use  of  this  method,  which  may  be  regarded  as  fundamental  condi- 
tions. 

a)  First  Condition  :  A  Fixed  Station. — The  first  requirement 
is  &  fixed  station,  from  which,  as  a  centre,  the  True  Bearing  of  the  object 
is  determined  ;  to  which  the  ship  may  be  held  either  directly  by  her 
anchor,  or  through  a  mooriug-buoy,  and  about  which  she  may  be  swung 
into  every  desired  Heading  round  the  compass-circle. 

This  requirement  is  usually  fulfilled  by  selecting  a  point  in  a  harbor 
or  roadstead,  round  which,  whatever  the  Heading  of  the  ship,  an  unob- 
structed view  of  the  distant  object  may  be  had  from  the  compass-posi- 
tion on  board.  The  selected  point,  as  a  centre  or  station,  is  sufficiently 
fixed  for  a  given  occasion,  by  casting  the  ship's  anchor  ;  but  it  is  some- 
times permanently  marked,  by  leaving  a  mooring-buoy  strongly  held 
by  a  mushroom  or  other  form  of  heavy  anchor. 

ft)  Second  Condition:  ITIeans  of  Swinging  the  Ship. — 

The  second  requirement  has  reference  to  the  means  of  swinging  the 
ship  upon  the  desired  Headings.  These  depend  somewhat  on  the  cir- 
cumstances of  the  location  and  the  resources  at  command. 

When  the  station  is  located  in  still  or  slack  water,  the  ship  may  be 
swung  by  means  of  warps,  using  for  this  purpose  kedges  and  hawsers  ; 
or  the  ship  may  be  pulled  about  with  great  advantage,  especially  if 
large,  by  a  steam-tug. 

When  the  location  is  in  a  stream  or  tide-way,  and  when  from  other 
circumstances  the  use  of  warps  maybe  inconvenient,  advantage  may 
be  taken  of  the  changing  positions  of  the  ship,  as  she  swings  at  her 
anchor  to  the  wind  and  tide,  to  obtain  a  sufficient  number  of  different 
Headings ;  and  these  may  sometimes  be  increased  in  number  and  variety 
with  the  aid  of  a  steam-tug. 

In  the  first  case,  it  is  generally  practicable  to  place  the  ship  with  her 
Head  upon  the  regular  points  by  compass — 32,  16,  8,  or  4,  as  may  be  de- 
sired $  but,  in  the  second  case,  it  is  generally  difficult,  if  not  impossible, 


METHOD    BY   DIRECT   BEARINGS. 


59 


to  do  this ;  and  the  only  resource  is  to  obtain  a  sufficient  number  of  dif- 
ferent Headings  round  the  whole  compass-circle,  even  if  they  are  not 
upon  the  regular  points. 

c)  Third  Condition:  A  limited  Parallax  of  Swing.— The 

third  requirement  is  that  the  distance  of  the  object  shall  be  sufficiently  great, 
in  comparison  with  the  radius  of  swing,  to  cause  no  appreciable  change  in 
the  Compass -Bear  ing  whatever  the  Position  or  Heading  of  the  ship  round  the 
station. 

This  requirement,  in  order  to  be 
clearly  understood,  may  need  a  little  de- 
tailed explanation.  Eef erring  to  the  fig- 
ure, let  O  represent  the  place  of  the 
distant  object;  S  the  centre  of  swing, 
or  station-centre  ;  C  the  position  of  the 
Standard  Compass  on  board. 

Then,  C  S  is  the  radius  of  siting, 
equal  to  the  horizontal  distance  be- 
tween the  compass  and  the  vertical  at 
the  mooring-anchor,  and  supposed  to 
remain  constantly  the  same,  or,  at  any 
rate,  not  greater  than  the  distance  fixed 
by  a  certain  length  of  cable ;  O  S  is  the 
distance  of  the  object,  which  is  sensibly 
the  same  as  O  C  in  all  positions  of  the 
ship  ;  and  A  S  C  is  the  angle  of  swing, 
which  may  have  any  value  from  o°  to 
1 80°,  on  each  side  of  the  zero-line,  O  A  S 
B,  through  the  object  and  centre  of  sta- 
tion. 

Now,  unless  the  distance  of  the  ob- 
ject be  quite  considerable  in  comparison 
to  the  radius  of  swing,  it  will  be  found 
that  the  direction  of  the  object  0  O,  as 
seen  from  the  compass  at  C,  may  be  sensibly  different  from  the  true  direc- 
tion S  O  as  seen  from  any  point  A  or  B  in  the  zero-line.  This  possible 
change  of  direction,  for  different  positions  of  the  ship,  is  measured  by 
the  Angle  O  C  O',  or  its  equal  COS,  and  is  called  the  Parallax  of  Swing. 
It  is  evidently  nothing  when  the  ship  is  in  the  zero-line,  as  when  headed 
towards  O  or  in  the  opposite  direction 5  and  it  increases  the  further  the 
ship  is  swung  from  that  line,  becoming  a  maximum  when  perpendicular 
to  it,  on  either  side. 

The  Compass-Bearings  of  the  object  are  accordingly  liable  to  an  Error, 
in  consequence  of  this  parallax,  which  is  called  the  Parallactic  Error  of 
Swing,  whose  magnitude  in  a  given  case  depends — 

1.  On  the  Distance-Ratio  of  the  Object;  that  is,  on  the  distance  of  the  ob- 
ject divided  by  the  radius  of  swing ;  and 

2.  On  the  Angle  of  Swing;  being  nothing  when  this  angle  is  o°  Or  180°, 
and  having  equal  maxima  when  the  angle  is  90°  and  270°. 


60  FINDING   THE   COMPASS-ERROR. 


68.  Process  of  Finding  Serial  Compass-Errors.— The 

Method  by  Bearings  of  a  Distant  Object  is  that  which  is,  perhaps,  most 
commonly  resorted  to  for  obtaining  a  series  of  Compass-Errors  round 
the  compass-circle.  It  may  be  conducted  in  the  following  manner : 

Everything  being  ready  and  the  requisite  observers  at  their  stations, 
swing  the  ship  so  as  to  bring  her  head  upon  the  nearest  point  by  Stand- 
ard Compass,  gently  checking  her  motion,  and  keeping  her  steady  upon 
the  point;  then,  take  the  Bearing  of  the  distant  object  with  the  Stand- 
ard Compass,  and  note  the  Heading  of  the  ship  with  the  same  compass. 
Note,  also,  the  Headings  with  the  steering-compasses,  and  the  Angle  of 
Heel  with  the  clinometer. 

Proceed  in  the  same  careful  manner  to  bring  the  ship's  Head  upon 
the  next  point,  and,  when  duly  stopped  and  steadied  there,  again  take 
the  Bearing  of  the  object  and  note  the  Heading  of  the  ship  with  the 
Standard  Compass,  noting  also  the  Heading  with  the  steering-compasses, 
and  the  Angle  of  Heel,  as  before ;  and  so  on,  point  after  point,  till  the 
Bearings  shall  have  been  taken  of  the  object,  with  the  corresponding 
Headings  of  the  ship  and  the  Angles  of  Heel,  for  every  point  of  the  com- 
pass round  to  the  point  of  beginning. 

The  operation,  as  thus  described,  may  be  performed  upon  each  of  the 
thirty-two  points,  sixteen  points,  or  eight  points  (preferably  in  that  case 
upon  the  octantal  points),  as  It  may  be  deemed  expedient  at  the  time. 

Otherwise,  if  the  circumstances  of  the  swinging  are  such  as  to  make 
it  inexpedient  to  attempt  obtaining  the  observations  on  the  regular  com- 
pass-points, proceed  rn  all  respects  in  the  same  manner,  except  that  of 
trying  to  bring  the  ship's  Head  upon  any  of  the  regular  points.  It  is, 
however,  desirable  in  this  case  to  distribute  the  observations  upon 
Headings,  approximately  at  an  equal  distance  apart,  round  the  compass- 
circle. 

In  either  case,  it  should  be  the  aim  of  the  observer  to  get  his  Bearings 
of  the  object  with  certainty  to  the  nearest  quarter  of  a  degree,  and  his 
Headings  to  the  nearest  eighth  of  a  point  or  nearest  whole  degree,  accord- 
ing as  he  is  making  his  observations  upon  the  compass-points  or  other- 
wise. 

The  record  of  the  observations,  which  should  be  completely  made  as 
the  work  proceeds,  may  be  kept  according  to  the  form  in  the  subjoined 
Examples. 


METHOD    BY    DIRECT   BEARINGS, 


61 


69.  Examples  of  the  Method  by  Direct  Bearing's. 

Ex.  1.     1872            :  At  ,  in  Lat.              .  ,  Long.  ,  Ship 

,  swung  for  a  set  of  Compass-Errors,  by  observations  of  a  distant  object,.  .. 

Distance  of  Object,  10  N.  miles  :  Radius  of  Swing,  200  feet. 

True  Bearing  of  object,  N  41°.  2  W. 
1 

ship's  Head. 

Direct  Bearing  of  Object. 

Error  of 
Standard 

By  Standard 
Compass. 

By  Standard 
Compass. 

True. 

Compass. 

o 

o 

o 

North. 

N  22.7  W 

N  41.2  W 

18.5  W 

N  by  E 

26.3 

• 

14.9 

NNE 

29.8 

• 

11.4 

NE  by  N 

30.8 

< 

10.4 

NE 

31.0 

' 

10.2 

NE  bv  E 

33-8 

• 

7-4 

ENE 

33-° 

< 

8.2 

E  by  N 

33-° 

' 

8.2 

East. 

31.8 

< 

94 

EbyS 

30.2 

* 

II.O 

ESE 

29.2 

• 

I2.O 

SEbyE 

28.2 

< 

I3.0 

SE 

25-5 

1 

J5-7 

SE  by  S 

24.0 

1 

17.2 

SSB 

22.8 

1 

18.4 

SbyE 

19.8 

1 

21.4 

South. 

18.0 

i 

23.2 

.. 

S  by  W 

*5-3 

' 

25-9 

ssw 

15.0 

< 

26.2 

SW  by  S 

13.0 

' 

28.2 

SW 

12.8- 

< 

28.4 

SW  by  W 

n.6 

' 

29.6 

wsw 

IO.O 

' 

31.2 

WbyS 

8.8 

' 

32-4 

/ 

West. 

8.4 

' 

32.8 

Wby  N 

9.0 

< 

32.2    • 

WNW 

9.8 

< 

3M 

NW  by  W 

11.4 

* 

29.8 

NW 

12.4 

< 

28.8 

NW  by  N 

14.8 

« 

26.4 

NNW 

18.0 

' 

23.2 

Nby  W 

19.6 

" 

• 

21.6  W 

62 


.  FINDING   THE   COMPASS-ERROR. 


Ex.  2.  —  1872,  February  29  :  At  a  station  ill  New  York  Bay,  Lat.  40°.5  N,  Long.  74°  W, 

U.  S.  Ship  CANANDAIGUA  (wood-built  screw-steamer).    Swung  for  a  set  of  Compass- 

Deviations,  by  Direct  Bearings. 

Objects  observed.                   Trite  Bearings.                  Distances.             Kad.  of  Swing. 

Sandy  Hook  L.  H.                S  880.g  E                 1.2  N.  M.               250  feet. 

Navesink  Lights.                  S  27  .1  E                 4.0      "                        " 

East  Beacon.                         N  60.  .9  E                 i.i      "                         " 

The  Mag.  Var.  wa.s  7°.6  W  at  the  place  and  date. 

Observations.                           Reductions. 

Ship's  Head. 

Direct  Bearing  of  Objects. 

Deviation  of 

the  Standard 

By  Standard 
Compass. 

By  Standard 
Compass. 

Magnetic. 

Compass. 

o 

o 

o 

North, 

S  81.5  E 

S  81.3  E 

0.2   E 

N  byE 

S  85.0  E 

81.3 

3-7 

NNE 

S  27.7  E 

S  19.5  E 

8.2 

.NE  byN 

S  31.2  E 

19-5 

11.7 

NE 

S  33.2  E 

19-5 

l3-7 

NEbvE 

N  84.0  E 

81.3 

14.7 

ENE 

S  36.8  E 

19-5 

17-3 

E  by  N 

S  37-o  E 

19-5 

17-5 

East. 

S  37.2  E 

S  19.5  E 

17-7 

E  bvS 

N  51.5  E 

N  68.5  E 

17.0 

ES'E 

N  85.2  E 

S  81.3  E 

"3-5 

SE  by  E 

N  87.0  E 

81.3 

11.7 

( 

SB 

N  88.8  E 

81.3 

9-9 

SE  by  S 

S  89.0  E 

81-3 

7-7 

SSE 

S  86.5  E 

81.3 

5-2 

S  by  E 

S  84.0  E 

81.3 

2.7  E 

South. 

S  81.5  E 

S  81.3  E 

0.2   E 

Sby  W 

S  79.0  E 

81.3 

2.3  W 

ssw 

S  75-5  E 

81.3 

5.8 

SW  by  S 

S  13.5  E 

S  19.5  E 

6.0 

sw 

S  n.2  -R 

19-5 

8-3 

SW  by  W 
\VSW 

S    8.8  E 
S    6.8  E 

19-5 
19-5 

10.7 
12.7 

Wby  S 

S    6.0  E 

19-5 

13-5 

West. 

S    5.0  E 

S  19.5  E 

H-5 

WbyN 

S  67.3  E 

S  81.3  E 

14.0 

WNW 

S    4.713 

S  19.5  E 

14.8 

NW  by  W 

S    5.7  E 

19-5 

13-8 

. 

NW 

S    8.2  E 

I9-S 

"•3 

NWbyN 
NNW 
Nby  W 

S    9.8  E 
S  75-5  E 
N  66.7  E 

!9-5 
S  81.3  E 
N  68.5  E 

It  w 

1.8  E 

• 

METHOD    BY   DIRECT    BEARINGS.  63 


70.  Dependence   to  be  placed  on  Results. — There  are 
two  distinct  sources  of  Error  which  pertain  to  the  Method  by  Bearings 
of  a  Distant  Object — 

First,  the  uncertainty  in  the  True  Bearing  of  the  object,  incident  to 
the  method  employed  in  finding  it;  and, 

Secondly,  inaccuracies  in  the  Compass-Bearings  of  the  object,  due  to 
the  Parallactic  Errors  of  Observation. 

The  first,  as  will  be  explained  hereafter,  should  never  exceed  the 
tenth  of  a  degree;  that  is,  ±o°.i  or  =L6'.  This  Error,  therefore,  being 
small  and  constant,  or  affecting  all  the  observations  alike,  may  gener- 
ally be  disregarded. 

The  second,  on  the  other  hand,  is  not  un frequently  a  much  more  seri- 
ous matter,  since  it  may  not  only  be  several  times  as  large  as  the  former, 
but  is  attended  with  the  complication  of  being  unequal  for  the  different 
Headings,  and  of  having  contrary  signs  on  opposite  sides  of  the  zero- 
line.  Nevertheless,  with  strict  attention  to  the  Distance-Eatio,  as 
will  presently  be  explained,  these  Errors  may  be  kept  within  admis- 
sible limits;  and,  besides,  they  may  always  be  estimated  and  applied 
as  corrections  of  the  observations,  whenever  it  is  certain  that  the 
swinging  is  done  throughout  with  a  taut  cable. 

71.  Tables  £.11  and  LI  SI  of  Parallactic  Errors:  Ex- 
amples.— Tables  LII  and  LIII  are  constructed  in  order  to  afford  the 
means  of  readily  estimating  these  Errors  whenever  desired.    They  re- 
quire no  explanation.    A  few  Examples  will  illustrate  their  use. 

Ex.  1.— The  Distance  of  the  object  is  7.4  iiaut.  miles,  the  Radius  of  Swing  150  feet, 
and  the  Angle  of  Swing  90° ;  What  is  the  Distance-Ratio,  and  what  the  greatest  Paral- 
lactic Error  I 

From  Tab.  LII  the  Dist.  Ratio  is  300,  and  from  Tab.  LIII  the  greatest  Parallactic 
Error  is  12',  or  o°.2. 

Ex.  2. — What  is  the  Distance-Ratio,  for  which  the  greatest  Parallactic  Error  shall 
not  exceed  6',  or  o°.i  ? 
From  Tab.  LIII  the  Dist.  Ratio  is  between  500  and  600. 

Ex.  3.— The  Distance  of  the  object  is  8  naut.  miles,  and  the  greatest  Parallactic 
Error  must  not  exceed  6' :  What  is  the  greatest  admissible  Radius  of  Swing  ? 
Am.  The  Dist.  Ratio  is  500,  and  the  Radius  of  Swing  must  not  exceed  100  feet. 

Ex.  4. — The  Distance  of  the  object  is  2  naut.  miles,  and  the  Radius  of  Swing  200 
feet :  WThat  is  the  greatest  Parallactic  Error  ? 
Ans.  i°.o. 

Ex.  5. — The  Radius  of  Swing  is  150  fest,  and  the  greatest  Parallactic  Error  is  not 
to  exceed  6',  or  o°.i  :  What  is  the  least  Distance  of  Object  that  will  be  required  ? 
Ans.  12  N.  M. 

73.  To  correct  the  Observations  for  Parallactic 
Errors, — It  will  be  the  work  of  but  a  few  moments,  and  it  may  some- 
times be  expedient,  to  correct  the  Observations  of  a  Distant  Object  for 
Parallactic  Error ;  or,  what  is  the  same  thing,  to  reduce  them  to  what 
they  would  have  been  if  they  had  been  made  at  the  centre,  or  on  the 
zero-line,  at  the  station. 


64  FINDING    THE    COMPASS-ERROR. 


Thus,  having  deduced  the  Compass-Errors  or  Deviations,  as  the  case 
may  be,  if  it  be  desired  to  correct  them  for  Parallax  of  Swing,  proceed 
in  the  following  manner : 

a)  Rule :  Case  of  Using  the  True  Bearing  of*  the 
Object. — Change  the  Compass-Headings  into  True  Headings,  by  ap- 
plying the  Compass-Errors  to  the  former,  to  the  Bight  when  these  Errors 
are  -£/,  to  the  Left  when  they  are  W. 

Next,  find  the  Angles  of  Swing  (68),  by  comparing  the  True  Headings 
of  the  ship  in  succession  with  the  True  Bearing  of  the  object;  setting 
down  the  Differences,  or  their  Supplements  when  greater  than  90°,  and 
marking  them  R.  or  Z/.,  according  as  the  True  Heading  is  Right  or  Left  of 
the  True  Bearing  of  the  object.1 

Lastly,  enter  Tab.  LIII  with  the  Distance-Ratio  and  the  Angles  of 
Swing,  and  take  out  the  corresponding  Parallactic  Errors ;  applying  to 
the  Eight  or  Left  of  the  Compass-Errors,  according  as  the  Angles  of 
Swing  are  L.  or  JR.  of  the  zero-line  (68)  or  True  Bearing  of  the  object 
from  the  station-centre. 

It)  Rule :  Case  of  11  sing  the  Magnetic  Bearing  of  the 
Object, — If  the  Magnetic  Bearing  of  the  object  be  used  instead  of  the 
True  Bearing,  and  the  Deviations  deduced  instead  of  the  Total  Errors, 
the  Rule  will  be  the  same  in  every  particular  as  the  preceding,  except 
in  substituting  therein  Magnetic  for  True,  and  Deviations  for  Compass- 
Errors. 

c)  Remark  I :  Observing  two  or  more  Objects.— If,  in- 
stead of  one  object,  two  or  more  be  employed  with  different  Bearings, 
True  or  Magnetic,  proceed  according  to  the  Rule,  only  using  the  proper 
Bearing  for  the  Compass  Error  or  Deviation  considered. 

d)  Remark  II :  Swinging  with  a  Taut  Cable.— It  must, 
of  course,  be  understood  that  these  corrections  can  only  be  reliably 
made  when  the  entire  swinging  is  done  with  a  taut  cable. 

e)  Remark  III:  Using  the  nearest  whole  Degree.— It 

is  sufficient,  in  these  corrections,  to  use  the  Bearings  and  Headings  to 
the  nearest  whole  degree. 

JThe  observer's  eye  being  supposed  at  the  centre  of  the  com  pass  and  looking  toward 
the  object. 


METHOD    BY    DIRECT    BEARINGS. 


65 


73.  Examples  of  Correcting  for  Parallactic  Errors. 

Ex.  1,—  The  Data  and  Results  of  Ex.  2  of  Art.  69:  The  True  Bearings  being  used 

and  Compass-Errors  deduced:  To  correct  them  for  Parallactic  Errors. 

Data  and  Results  of  Ex.  2.           Correcting:  for  Paral.  Errors. 

Ship's  Head 
by  Standard 
Compass. 

True  Bear- 
Ings  of  Ob- 
jects. 

Error  of 
Standard 
Compass. 

Ship's 
Head, 
True. 

Angles  of 
Swing. 

Parallactic 
Errors. 

Corrected 
Compass- 
Errors. 

North. 

S  89  E 

o 

7.4  W 

N    7  W 

82  L 

o 

1.9 

5-5  W 

NbvE 

" 

3-9  W 

7E 

84 

1.9 

2.0    W 

NNE 

S  27  E 

0.6  E 

23 

5° 

0.9  E 

NE  bv  N 

" 

4.1 

38 

65 

0.3 

4.4 

NE 

" 

6.1 

78 

0.4 

6.5 

NE  by  E 

S  89  E 

7-1 

63 

28 

I.O 

8.1 

ENE 

S  27  E 

9-7 

77 

76 

0.4 

IO.I 

E  by  N 

IO.O 

89 

64 

10.3 

East. 

S  27  E 

10.1  E 

S  80  E 

53  L 

o-3 

10.4  E 

E  bv  S 

N6i  E 

9-4 

69 

50  R  ' 

1.5 

7-9 

ESE 

S  89  E 

5-9 

62 

27 

0.9 

5-° 

SE  bv  E 

4.1 

52 

37 

.1 

3-° 

SE 

2-3 

43 

46 

•4 

0.9  E 

SE  by  S 

o.i  E 

34 

55 

.6 

1.5  W 

8SE 

2.4  W 

25 

64 

•  7 

4.1 

Sby  E 

4-9 

16 

73 

.8 

6-7 

South. 

S  89  E 

7-4  W 

S    7E 

81  R 

•9 

9-3  W 

S  by  W 

9-9 

i  W 

90 

•9 

11.8 

ssw 

13-4 

9 

82 

•9 

15-3 

SW  bv  S 

S  27  E 

13.6 

20 

47 

13-9 

SW 

29 

56 

0.3 

16.2 

SW  bv  W 

18.3 

38 

65 

0.3 

18.6 

WSW 

20.3 

47 

74 

0.4 

20.7 

W  by  S 

21.  1 

58 

85 

0.4 

21.5 

West. 

S  27  E 

22.1 

S  68  W 

85  R 

0.4 

22.5  W 

W  by  N 

S  89  E 

21.6 

80 

ii 

0.4 

22.O 

WNW 

S  27  E 

22.4 

N  90  W 

63 

22.7 

NW  bv  W 

" 

21.4 

78 

51 

0.3 

21.7 

NW 

" 

18.9 

64 

37 

O.2 

I9.I 

NW  bv  N 

" 

17-3 

51 

24  R 

O.2 

NNW 

S  89  E 

13-4 

36 

53  L 

1.6 

ii.S 

N  by  W 

N6i  E 

5.8  W 

i7 

78 

1.9 

3-9 

This  Example  supposes  the  Compass-Errors  instead  of  the  Deviations  to  have  been 

found.      It  illustrates  the  more  complicated  case  of  several  objects  being  observed 

instead  of  one.     The  procedure  is  entirely  the  same. 

/Al*W  T  T"  T5  T)    d  T  f  Vtt 

fm$              A-*  XH 

66 


FINDING    THE    COMPASS-ERROR. 


Ex.  2.  —  The  Data  and  Results  of  Ex.  2  of  Art.  69:  To  correct  tbo  Deviations  for 

Parallactic  Errors. 

Data  and  Results  tff  Ex.  2.           Correcting  for  Para!.  Errors. 

Ship's  Head 
by  Standard 
Compass. 

Magnetic 
Bearings  of 
Objects. 

Deviation 
of  Standard 
Compass. 

Ship's 
Head, 
Magnetic. 

Angles  of 
Suing. 

Paralhutic 
Errors. 

Corrected 
Compass- 
Deviations. 

North. 

S  8°i  E 

o 
0.2   E 

North. 

O' 

81  L 

o 

1.9 

o 
2.1    E 

N  bv  E 

.    81 

3-7 

N  15  E 

84 

1.9 

5.6 

NNE 

S  IQ  E 

8.2 

31 

50 

o-3 

.8-5 

NE  by  N 

19 

11.7 

45 

64 

°-3 

12.0 

NE 

19 

J3-7 

59' 

78 

0.4 

I4.I 

NE  by  E 

S  Si  E 

14.7 

7i 

28 

I.O 

15-7 

ENE 

19 

J7-3 

85 

76 

0.4 

177 

E  by  N 

19 

17-5 

S  84  E 

65 

0-3 

17.9 

East. 

S  19  E 

17-7 

S  72  E 

53  L 

°-3 

1  8.0  E 

E  by  S 

N69  E 

17.0 

62 

49  R 

i-5 

15-5 

ESE 

S  81  E 

J3-5 

54 

27 

0.9 

12.6 

SE  by  E 

81 

11.  7 

44 

37 

.1 

10.6 

SE 

81 

9-9 

35 

46 

•4 

8.5 

SE  by  E 

8r 

7-7 

26 

55 

.6 

6.1 

SSE 

81 

5-2 

17 

64 

•  7 

3-5 

Sby  E 

81 

2.7 

8 

73 

.8 

0.9  E 

South. 

S  81  E 

0.2    E 

South. 

81  R 

•9 

1.7  W 

S  bv  W 

81 

2.3  W 

S    9W 

90 

•9 

4.2 

ssvv 

81 

5.8 

17 

82 

•9 

7-7 

SW  by  S 

S  19  E 

6.0 

28 

47 

°-3 

6-3 

SW 

19 

8-3 

37 

56 

o-3 

8.6 

SW  by  W 

19 

10.7 

45 

64 

°-3 

II.O 

WSW 

19 

12.7 

55 

74 

0.4 

J3-1 

W  byS 

19 

13-5 

65 

84 

0.4 

13-9 

West. 

S  19  E 

14-5 

S  76  W 

85  R 

0.4 

14.9  W 

WbvN 

S  81  E 

14.0 

87 

12 

0.4 

14.4 

WNW 

S  19  E 

14.8 

N82  W 

63 

°-3 

i5-i 

NW  by  W 

19 

13.8 

70 

51 

o-3 

14.1 

NW 

J9 

"•3 

56 

37 

O.2 

n-5 

NW  bv  N 

*9 

9-7 

43 

24  R 

0.2 

9-9 

NNW 

S  81  E 

5.8  W 

28 

53  L 

1.6 

4.2  W 

Nby  W 

N69  E 

1.8  E 

9 

78 

1.9 

3-7  E 

It  will  be  seen  that,  the  Angles  of  Swing  are  the  same,  as  they  should  lu>,  in  these 

two  Cases  of  the  same  Example.     The  Parallactic  Errors  are  consequently  the  same. 

It  is  quite  evident  that  Ex.  2,  Art.  69,  should  be  corrected  for  Parallactic  Errors. 

METHOD    BY    DIRECT   BEARINGS.  67 


74.  Limiting    Distance- Ratios. — From    an    inspection  of 
Tables  LII,  LIII,  and  the  Examples  of  Art.  71,  the  need  of  caution  in 
the  use  of  this  method  will  be  evident. 

Assuming  that  we  are  content  to  admit  a  maximum  Parallactic  Error 
of  o°.i,  or  6',  the  Distance-Ratio  will  have  to  be  not  less  than  500;  and, 
with  this  as  a  limit,  it  will  be  seen  that,  even  with  a  Eadius  of  Swing  no 
greater  than  100  feet,  we  shall  require  the  distance  of  the  object  to  be  not 
less  than  8  nautical  miles. 

If,  however,  in  a  given  case,  the  Distance-Ratio  be  much  less  than  500, 
and  we' can  neither  find  an  available  object  farther  off,  nor  in  any  prac- 
ticable manner  materially  diminish  the  Radius  of  Swing,  we  shall  then 
have  to  consider — 

First,  whether  we  shall  still  proceed  and  correct  the  observations  for 
Parallactic  Errors ;  or, 

Secondly,  whether,  if  this  be  impracticable  (72,  <Z),  we  can  afford  to 
disregard  these  Errors,  under  the  circumstances  of  the  case ;  or, 

Thirdly,  whether  we  should  not  preferably  resort  to  one  of  the  other 
methods. 

75.  Different  methods  or  Finding'  the  True  Bear- 
ing1 of  the  Object. — In  using  permanent  stations  for  the  observation 
of  Compass-Errors  by  this  method,  the  True  Bearing  of  the  object  is  com- 
monly known  from  a  previous  determination.     Whenever  unknown,  as 
when  establishing  a  new  station,  temporary  or  otherwise,  the  True  Bear- 
ing will  be  required,  and  it  may  be  found  by  one  of  the  methods  named 
below : 

1.  By  the  Geographical  Position. 

2.  By  an  Astronomical  Bearing. 

3.  By  an  Azimuth  of  the  Vertical  Circle. 

In  describing  these  methods,  we  shall  have  to  keep  in  mind  that, 
among  the  requirements  of  the  station  (67),  it  must  be  at  an  anchorage, 
and  consequently  the  observations  for  the  True  Bearing  must  be  made 
from  the  deck  of  the  ship,  although  it  may  be  possible  in  most  cases  to 
select  a  smooth  sea,  with  other  favorable  circumstances. 

76.  Finding  the  True  Bearing  by  the  Geographical 
Position. — When  the  Geographical  Position  of  the  object  is  reliably 
known,  the  True  Bearing  and  Distance  are  easily  determined  in  the 
following  manner  : 

Rule: — Deduce  the  Co-Latitude  of  the  object  and  the  Co-Latitude  of 
the  station  ;  also,  the  Difference  of  Longitudes  between  the  object  and 
station.  Then  proceed  according  to  the  method  of  Time-Azimuths  (15) ; 
treating  the  Co-Latitude  of  the  object  like  the  Pol.  Dist.  and  the  Diff. 
Longitudes  like  the  Hour- Angle  of  a  celestial  body;  following  the  Rule 
^19),  either  by  computation  (22)  or  inspection  (26),  in  every  particular. 

Having  found  the  True  Bearing,  enter  a  Traverse-Table  with  this 
Bearing  as  Course  and  the  Diff.  of  the  two  Latitudes  as  D.  Lat.  and 
take  out  the  Distance  of  the  object. 


68  FINDING    THE    COMPASS-ERROR.' 


77.  E\ample§  of*  Finding  the  True  Bearing  by  the 
Geographical  Position. 

Ex.  1. — The  respective  Positions  of  the  station  and  object  are  as  below  stated  : 
What  is  the  True  Bearing  of  the  object  from  the  station,  and  what  is  its  Distance  in 
nautical  miles? 

Lat.  Long. 

Station  in  Harbor  of  Yokohama    35°  26'. o  N     139°  39'.o  E 

Distant  object  (Fusi-Yaina)  35    21  .5  N     138   41  .oE 

Co-Lat.  54°  38'-5  object  With  Diff.  Lat.  40.5  and  True 

Co-Lat.  54   34.0  station"  Bearing  84°.8,  as  Course,  we  take 

Diff.  4.5  from   Traverse  -  Table   the  Dis- 

y2  Diff.  2 .2  sin      6.8159    cos  o.oooo          tance  50  N.  miles. 

y2  Sum  54  36.5  cosec  0.0887    8600.2371 

Yz  D.  Long.          28^  cot     2.0776    cot  2.0776 
tan     8.9822    tan  2.3147 

T.  BEAK.    N95°i2'W:X      5°  29'     Y  89°  43' 
or,      S  84   48  W 

Ex.  2. — The  Positions  of  the  station  and  object  are  as  below  stated :  It  is  required 
to  find  the  True  Bearing  and  Distance  of  the  object  from  the  station. 

Lat.  Long. 

Station  at  sea,  in  60°    6'  N     142°  50'  W 

Object,  Mt.  St.  Elias,  in  60    18  N     140   52  W 

Co-Lat.  29°  42'  object                                                 With  Diff.  Lat.  12',  True  Bear- 

Co-Lat.  29   54  station                                          ing  77°.6,  as  Course,  the  Trav- 

Diff.                        12  erse-Table gives  for  Distance  57 

Yz  Diff.                     6  sin      7.2419    cos  o.oooo  N.  miles. 

}/2  Sum  29    48  cosec  0.3037    sec  0.0616 

}/2  D.  Long.           59  cot      1.7654    cot  1.7654 
tan     9.3110    tan  1.8270 

T.  BEAR.  N  77°  35'  E  :  X  n°  34'        Y  89°  9' 

78.  By  an  Astronomical  Bearing,— When  the  Geograph- 
ical Position  of  tlie  object  is  not  reliably  known,  the  preceding  method 
of  finding  the  True  Bearing  is  no  longer  available.     In  such  a  case, 
among  other  means,  the  method  by  an  Astronomical  Bearing  may  be 
used.    This  consists  in  finding  the  Difference  of  Bearings  between  the 
object  and  some  celestial  body,  whose  True  Azimuth  is  determined  at 
the  same  time  by  one  of  the  methods  of  Section  (/). 

a)  Bule  :  To  Ulake  the  Observation.— Measure  the  Angular 
Distance  with  a  sextant  between  the  denned  point  of  the  object  and  the 
Sun's  lirnb  ;  have  a  second  observer  measure  the  Altitude  of  the  Sun  at 
the  same  moment,  with  a  sextant  or  octant;  and  note  the  Time  with  a 
Com  paring- Watch,  whose  error  on  Local  Time,  Apparent  or  Mean,  is 
reliably  known. 

Otherwise,  in  the  absence  of  the  second  observer,  first,  measure  the 
Altitude  of  the  Sun  ;  next,  the  Distance  between  the  defined  point  of  the 
object  and  the  Sun  ;  then,  a  second  Altitude  of  the  Sun  ;  each  observa- 
tion being  taken  without  unnecessary  delay,  and  the  W.  Time  of  each 
being  noted. 


METHOD    BY    DIRECT    BEARINGS.  69 


In  either  Case,  take  tlie  Altitude  of  the  defined  point  on  the  object 
above  the  Sea  or  Shore  Horizon,  as  the  circumstances  may  require. 
Note,  also,  whether  the  object  is  Eight  or  Left  of  the  observed  celestial 
body. 

b]  Rule:  To  ]?Iake  the  Computation.— First,  find  the  True 
Az.  of  the  celestial  body,  either  by  the  Time-Method  (15)  or  by  the  Alti- 
tude-Method (41),  according  as  the  Time  or  Altitude  is  the  most  reliable, 
or  as  may  be  preferred  for  any  other  reason. 

Secondly,  find  the  Az.  Difference,  as  follows :  Change  the  Observed 
Altitudes  of  the  celestial  body  and  terrestrial  object  into  Apparent  Alti- 
tudes by  correcting  them  for  Index-Error  of  the  sextant,  and  for  Dip  ;l 
also,  for  the  Semi-Diameter,  if  the  Sun  or  Moon  be  the  body  observed. 

Change  the  Observed  Angular  Distance  into  Apparent  Ang.  Distance, 
by  correctibgit  for  Index-Error,  and  for  Semi-Diameter,  whenever  neces- 
sary. 

Add  together  the  Ap.  Dist.  and  the  two  Ap.  Alts. ;  take  the  Difference 
between  their  Half-Sum  and  the  Ap.  Dist.  Then  add  together  the 
secants  of  the  two  Ap.  Alts.,  the  cosine  of  the  Half-Sum,  and  the  cosine 
of  the  Difference.  The  Half-Sum  of  these  four  Logs,  rejecting  tens 
from  their  indices,  will  be  the  cosine  of  half  the  Az.  Diff.,  which  take 
out  and  double,  and  the  result  will  be  the  required  Az.  Diff. 

There  will  now  be  two  Cases  to  consider,  according  as  the  Az.  Diff., 
reckoned  from  the  celestial  body,  is  in  the  same  sense  as  the  True  Az., 
or  in  the  opposite  sense. 

First  Case :  The  Az.  Diff.  in  the  same  sense. — Take  the  Sum  of  the 

True  Az.  and  Az.  Diff.,  and  the  result  will  be  the  True  Bearing  of  the 
object ;  which,  if  less  than  180°,  will  have  the  same  name  as  that  of  the 
True  Az.;  but,  if  greater  than  180°,  subtract  from  360°,  and  change  the 
meridional  reference. 

Second  Case;  The  Az.  Diff.  in  the  opposite  sense. — Take  the  Differ- 
ence of  the  True  Az.  and  Az.  Diff.,  and  the  result  will  be  the  True  Bear- 
ing  of  the  object ;  which  will  have  the  same  or  different  meridional  refer- 
ence, according  as  the  True  Az.  is  greater  or  less  than  the  Az.  Diff. 

79.  Examples  of  Finding-  the  True  Bearing*  by  an 
Astronomical  Rearing. 

Ex.  1.— 1870,  May  20 :  At  anchor,  in  Lat.  40°  30'  N,  Long.  74°  o'  W,  about  8Xh  A.  M., 
made  the  following  observations  for  finding  the  True  Bearing  of  a  distant  light-house: 

T.  by  Watch.  Obs.  Alt.  0  Ang.  Dist.  Lt.2  Obs.  Alt.  of  Lt.2 

gh  I0m  50s  38°  !g'  ,5" 

12    50  0|  80°  5'  20"  R.  50  10'  35" 

14    50  39     3  20 

Watch  slow  of  ship  A.  T.  5™  10*;  Ht.  of  eye,  18  feet;  Ind.  Corr.  —  2'  2$". 

1  When,  as  may  frequently  happen  in  the  nse  of  this  method,  the  Shore,  instead  of 
the  proper  Sea  Horizon,  is  observed,  the  usual  correction  for  a  "Shore-Dip"  must  be 
used. 

3 Top  of  lantern. 


70  FINDING    THE    COMPASS-ERROR. 


Preparation  of  Data. 

Mid.  Ship.  A.T.  i9''2oh  18™  0's  Dec.         X  19°  59'.7\+y"  Mid.  Obs.  Alt.  38°  4o'.8  Ind.  Corr.    -  2'.4 

Loug.  in  T.           +     4    56  lied"   for  ih.2     -f      0.6      37.2  Alt.  Corr.  -f-     9.4  Dip.             —4.0 

Gr.  date  May      20      i    14  Red.  Dec.            20     0.3  A  p.  Alt.  -0-  38    50.2  Semi-D.       4-15.8 

or,                       1.2  Pol.  Dist.           70     o  Alt.  Corr.   +9.4 

Mid.  Ap.  T.  8h  18"                  Obs.  Aug.  Dist.  80°  5'.5  Ubs.  Alt.  of  Lt.  5°  io'.6 

Mid.  H.  A.    3  42  E                 Ind.  Cun.  2.4  Alt,  Corr.             -6.4 

Semi-D.  -f    15  .8  Ap.  Alt.  Lt.        5      4 .2 

Ap.  Aug.  Dist,  3o   18  .9 

Computation  of  True  Az.  of  Sun.  Computation  of  Diff.  Az. 

Pol.  Dist.    70°    o'                                                              Ap.  (list.  80°  iS'.g 

Co-Lat.       49    30                                                            Ap.  Alt.  Lt.  5      4.2    sec    0.0017 

Diff.            20    30                                                            Ap.  Alt.  0-  38    50.2     sec    0.1085 

]4.  Diff.        10    15     siu  9  2503    cos  9.9930              Sum  124    13 .3 

%  Sum       59    45     cosec  0.0636    sec  0.2978              ^  Sum  62     6 .6    coa    9.6700 

Yz  H.  A.       ih  51™  cot  0.2789    cot  0.2789              Hem.  18    12.3    cos   9.9777 

tan  9.5928     tan  0.5697  19-7579 

T.  Az.      N    g6°i9'E:X  21°  23'     Y  74°  56'              ^  D.  Az.  40    50        cos    9.8789 

Diff.  Az.         8 1     40  R.                                                           Diff.  Az.  81    40 
T.  BEAR.  N  177    59  E 
or,        S      2      i  E 

Without  multiplying  examples  in  finding  the  True  Azimuth  of  a  celes- 
tial body,  already  fully  exemplified,  the  following  three  Examples  illus- 
trate the  Rule  for  deducing  the  True  Bearing  from  the  True  Az.  and  Az. 
Diff. 

Ex.  2.— Suppose  the  True  Az.  N  125°  E  aud  the  Az.  Diff.  87°  to  right. 
Then,  True  Az.  N  125°  E 

Diff.  Az.        87   R.  aud  therefore  iu  same  sense. 

212    greater  than  180. 

T.  BEAK.  N  148   W  with  the  meridional  reference  changed  from  E  to  W,  accord- 
ing to  the  2d  alternative  of  the  First  Case. 

Ex.  3.— Suppose  the  True  Az.  N  76°  W  and  the  Az.  Diff.  53°  to  right. 

Then,  True  Az.  N  76°  W 

53   R.  in  the  opposite  sense  and  less  than  True  Az. 
T.  BEAR.  N  23   W 

Ex.  4.— Suppose  the  True  Az.  N  61°  E  and  Az.  Diff.  77°  to  left. 

Then,  True  Az.  N  610  E 

77   L.  in  the  opposite  sense  and  greater  than  True  Az. 
T.  BEAR.  N  16   W 

8O.  By  an  Azimuth  of  the  Vertical  Circle.— A  conven- 
ient and  simple  means,  which,  with  care  in  the  observation,  may  be  em- 
ployed with  the  requisite  precision,  is  to  note  the  Time  when  the  Sun  or 
some  other  celestial  body  crosses  the  Vertical  Circle  of  the  object,  above 
it,  and  find  the  True- Azimuth  of  the  body,  which  will  be  that  of  the. 
terrestrial  object. 

a)  Rule:  To  take  the  Observation.— Be  ready  with  a  plumb- 
line,  so  held  as  to  cover  the  point  of  observation  on  the  object,  a  lew 
moments  in  advance  of  its  expected  transit  by  a  recognized  celestial 
body;  also,  with  an  assistant  to  note  the  time;  proceeding  as  follows: 

i.  Tfthe  Body  be  the  Sun  or  the  Full  Moon,  note  the  Times,  with  a  watch, 
of  the  first  and  second  contact,  or  the  instants  when  the  inner  limb 


METHOD    BY    DIRECT    BEARINGS.  71 


touches  the  plumb-line  and  the  outer  limb  leaves  it,  and  take  the  Mean 
of  the  two  Times  as  the  time  of  crossing  by  the  centre  of  the  body. 

2.  If  the  Body  be  a  Planet  or  Star,  note  the  Time  at  the  instant  the 
body  crosses  the  plumb-line. 

&)  Rule:    To  find  the  True  Bearing*  of  the  Object.— 

By  the  method  of  Time-  Azimuths  (19)  find  the  True  Azimuth  of  the 
celestial  body  for  the  Time  at  which  it  crosses  the  Vertical  Circle  of  the 
object,  and  the  result  will  be  the  required  True  Bearing. 

81.    Examples  of*  Finding  the  True  Bearing  by  an 
Azimuth  of"  fliie  'Vertical  Circle. 


Ex.  1.  —  1870,  July  23  :  At  anchor,  in  New  York  Bay,  in  Lat.  40°  27'.  6  N,  Long.  74° 
i'-5  W,  at  6h  i6m  56s  LOG.  Ap.  Time,  P.  M.,  the  Sun's  centre  crossed  the  Vertical  Circle 
above  the  Red  Bank  Light:  Required,  the  True  Bearing  of  the  Light. 

LOG.  Ap.  T.      23d  6'1  16™  56*  Q's  Dec.  N  20°   5x.i|  —  o'.5         H.  A.  W. 

Long.  +    4    56       6  Rednforiib.2  5.6       5.5         6h  i6m  56" 

Gr.  date  July  23  n     13       2  Red.  Dec.  19   59.5 

1  1.  2  Pol.  Dist.  70     0.5 

Pol.  Dist.  700  o'-5 
Co-Lat.  49  32  .4 
Diff.  20  28.1 

)4  Diff.          10    14          sin  9.2496    cos  9.9930 

)4  Sum         59  46          cosec  0.0635    sec  0-2981 

yz  H.  A.  3h  8m  283          cot  9.9679    cot  9.9679 

tan  9.2810    tan  0.2590 

T.  BEAR.  N  71°  58'  W:   X  io«  49'    Y    61°  9' 

82.  Remarits  relative  to  Finding*  the  True  Bearing 
of  the  Ohject.  —  The  first  method  of  finding  the  True  Bearing  pre- 
sents the  advantage  of  requiring  no  observations  of  Data;  so  that,  if 
the  Geographical  Positions  of  the  object  and  station  are  reliably  known, 
this  method  is  simple  and  convenient.  Still,  it  should  not  be  relied  on 
implicitly  for  precision  of  result.  The  small  Polar  Angle  (Diff.  Longi- 
tudes) arid  correspondingly  small  Distance  Arc  are  unfavorable  condi- 
tions (60,  b)  •  but,  on  the  other  hand,  the  difference  between  the  two 
Latitudes  is  always  so  small,  that  the  Position-  Angle  of  the  object  never 
differs  much  from  90°,  which  is  a  favorable  condition  (60,  6).  Conse- 
quently, Errors  of  the  Data  do  not  greatly  influence  the  result.  Thus, 
if  the  two  Latitudes  and  the  Difference  of  Longitudes  may  be  depended 
on  within  one  minute  of  arc,  the  True  Bearing  may  generally  be  de- 
pended on  at  least  within  two-tenths  of  a  degree. 

The  second  method,  though  attended  with  greater  labor  of  observa- 
tion and  computation,  is  rather  more  reliable  than  the  first  in  precision 
of  result.  Thus,  if  each  of  the  Data  can  be  depended  on  within  one  min- 
ute of  arc,  the  True  Azimuth  of  the  celestial  body  and  the  Azimuth-Differ- 
ence between  the  body  and  object,  may  in  general  be  depended  on,  each 
to  the  nearest  tenth  of  a  degree  (63);  so  that  the  True  Bearing  of  the 
object  should  be  depended  on  within  one-tenth  of  a  degree.  But  even 
this  precision,  not  very  close,  can  only  be  assured  of,  in  observations 
from  the  deck  of  a  ship,  with  favorable  circumstances  and  special  care 


72  FINDING   THE    COMPASS-ERROR. 


in  the  observations.  The  terrestrial  object  must  have  a  well-defined 
point  of  observation,  which  may  always  be  seized  with  certainty  when- 
ever referred  to.1 

The  third  method,  so  far  as  the  True  Azimuth  of  the  Vertical  Circle  is 
concerned,  should  also  be  depended  on  to  the  nearest  tenth  of  a  degree 
(63).  The  Error  of  Observation,  in  noting  the  time  when  the  centre  of 
the  celestial  body  crosses  the  plumb-line,  will  be  different  according  to 
circumstances  and  the  care  of  the  observer.  Its  effect  should  not,  how- 
ever, exceed  the  proper  Error  of  Azimuth ;  so  that,  on  the  whole,  the 
True  Bearing  of  the  object  should  be  depended  on  within  one- tenth  of  a 
degree. 

B.  METHOD  BY  ALLIGNMENTS. 

83.  Fundamental  Idea. — From  the  outer  approaches  to  almost 
every  harbor  or  roadstead  of  importance,  certain  pairs  of  prominent 
objects  may  be  sighted,  such  as  two  light-houses,  two  towers,  etc.,  whose 
Allignments,  or  Lines  of  True  Bearing,  have  been  determined  and  laid  down 
on  the  published  charts  and  plans  of  these  harbors  for  the  convenience 
of  Navigators. 

These  Allignments,  among  their  different  uses,  furnish  convenient 
means  for  finding  the  Compass-Error,  whenever  steaming  or  sailing 
across  them.  The  difference  between  the  observed  Compass-Bearing  of 
the  Allignment  and  its  True  Bearing  per  chart  is  the  Compass-Error. 

84.  Process  or  Finding  Serial  Compass-Errors.— -The 

use  of  Alignments  for  this  purpose  is  very  simple.  It  is  only  neces- 
sary to  observe  the  Compass-Bearing  of  the  Alligument  as  the  compass 
comes  into  the  line,  noting  at  the  same  time  the  Heading  of  the  ship 
with  the  same  compass.- 

Allignments  are  sometimes  employed  in  finding  Serial  Compass-Errors, 
the  operations  being  conducted  in  tbe  following  manner : 

Everything  being  ready,  and  the  observers  at  their  stations,  head  the 
ship  for  a  alow  run  if  under  sail,  or  bring  the  ship  by  any  convenient 
evolution,  if  under  steam,  across  the  Allignmeut ;  steady  the  ship  ;  and, 
at  the  precise  moment  when  the  observer  at  the  Standard  Compass 
gets  both  objects  in  his  line  of  sight,  note  the  bearing  of  the  Allign- 
ment ;  note  also  the  ship's  Head  by  the  Standard  and  Steering  Com- 
passes, and  note  the  Angle  of  Heel  by  clinometer. 

Proceed  in  the  same  manner,  upon  a  second  tack  under  sail  or  with  a 
similar  evolution  under  steam,  so  as  to  place  the  ship  again  across  the 
Allignment,  changing  the  Heading  by  the  desired  amount ;  and,  again 

llt  will  be  noticed  that  this  method  is  supposed  in  the  text  to  be  used  only  in  find- 
ing the  True  Bearing  of  a  terrestrial  object,  for  observations  of  Serial  Compass-Errors, 
at  a  fixed  station.  The  "Astronomical  Bearing"  is  commonly  given  in  the  books  on 
Navigation  as  a  method  of  finding  the  "Variation  of  the  Compass,"  or  Compass-Error  ; 
but  it  is  difficult  to  conceive  of  a  case  in  which  the  Navigator  need  ever  resort  to  so 
cumbrous  means  of  doing  what  would  be  better  done  by  a  small  part  of  the  process. 


METHOD    BY    ALLIGNMENTS. 


73 


steadying,  note  the  Bearing,  the  Heading,  and  Angle  of  Heel  as  be- 
fore. 

Continue  the  operation  by  a  succession  of  sailing-tacks  or  steam-evo- 
lutions, crossing  the  Allignment  each  time  under  a  different  Heading, 
and  making  the  corresponding  observations,  until  a  sufficient  number 
shall  have  been  obtained. 


85.  Examples  of  the  method  toy  Alligmiieiits. 

Ex.  1, — Harbor  of  Toulon  :l  By  Compass-Bearings  of  an  Alligument  of  Cape  Brun 
and  Fort  Faron,  made  the  following  observations  for  Compass-Error  : 
True  Bearing  of  Allignment  N  12°  30'  W. 
Distance  between  objects  2.2  N.  miles. 
Distance  (average)  of  tbe  ship  from  the  outer  object — not  stated, 


Bearing  of  Allignment. 

Ship's  Head 

Compass- 

Stand.  Comp. 

Error. 

stand,  romp. 

True. 

N    6  E 

N     o  15  W 

N  i°2  30  W 

12°  15  W 

N  35  E 

N  15  30  W 

3    oE 

N  65  E 

N  20    o  W 

7  30  E 

S  60  E 

N    83oW 

4   oW 

S  50  E 

N    6    oW 

6  30  W 

S  23  E 

N    2  45  E 

i5i5W 

S  24  W 

N  14  15  E 

2645  W 

S  62  W 

N  20  45  E 

33  15  W 

N  85  W 

N  26  30  E 

39    oW 

N  70  W 

N  26  45  E 

39,i,i  W 

N  20  W 

N  14    o  E 

26  30  W 

The  order  in  which  the  observations  are  set  down  is  not  necessarily 
that  in  which  they  were  made.  It  is  obvious  that  the  Headings  are  not 
well  distributed  round  the  compass-circle ;  but  this  is  not  always  possi- 
ble, when  the  observations  are  made  by  tacking  across  the  Allignment 
under  sail.  But  even  in  this  case  it  will  be  possible  to  obtain  a  tolera- 
ble Table  of  Errors  in  the  manner  to  be  explained  at  a  later  stage. 


1  This  Example  is  from  Traiti  de  Deviation  et  de  .Emulation  des  Campos,  p.  62  :  par  E. 
GIQUEL,  Paris,  1868. 

The  observations  are  said  to  have  been  made  on  board  the  Floride,  a  sailing-ship,  on 
a  succession  of  tacks  across  the  Allignment. 


10 


74 


FINDING   THE   COMPASS-ERROR. 


86*  Dependence  to  be  placed  on  Results. — There  are  two 
sources  of  Error  in  the  Method  by  Alignments: 

First-,  unless  the  recorded  True  Bearing  of  the  Alligument  has  been 
determined  by  geodetical  means,  it  is  liable  to  be  inaccurate  ;  it  may  be 
but  a  few  minutes,  though  it  is  quite  possible  to  be  several  tenths  of  a 
degree,  in  Error. 

Secondly,  there  is  a  liability  to  Errors  of  Parallax  in  observing  the  Com- 
pass-Bearings. In  general,  the  liability  to  these  Errors  is  increased  in 
proportion  as  the  distance  of  the  ship  from  the  first  or  outer  object  is 
greater  than  the  distance  between  the  objects.  Serious  Errors  of  this 
kind  can  only  be  avoided  by  great  care  in  bringing  the  ship  to  a  perfect 
"  steady,"  when  across  the  line,  and  in  getting  the  Bearing  at  the  mo- 
ment when  the  compass  is  precisely  on  that  line. 

Remark. — Except  as  a  convenient .  means  of  testing  the  com- 
passes on  a  certain  Heading,  when  running  across  an  Allignment,  this 
method  is  not  likely  to  be  much  used,  in  preference  to  the  other  methods. 
And  it  should  hardly  be  resorted  to  in  preference  to  the  method  by 
Solar  Azimuths,  on  any  occasion  of  finding  Serial  Compass-Errors. 


C.  METHOD  BY  RECIPROCAL  BEARINGS. 


87.  Preliminary  Explanations.— The  Method  by  Reciprocal 
Bearings  consists  in  the  use  of  two  instruments,  with  which  Simultaneous 
Bearings  are  taken,  each  of  the  other;  the  Standard  Compass  on  board 
being  used  for  the  Bearings  of  a  station  on  shore,  at  which  the  second 


instrument^  used  for  the  corresponding  Bearings  of  the  compass-station 
on  board. 


METHOD  BY  RECIPROCAL  BEARINGS.  75 


This  method  differs  in  an  important  particular  from  the  Method  by 
Bearings  of  a  Distant  Object:  There  is  obviously  no  necessity  for 
a  fixed  station,  round  which  the  ship  is  to  be  swung.  Each  pair  of 
Reciprocal  Bearings  by  a  direct  comparison  furnishes  the  Compass- 
Error  for  a  certain  Heading  of  the  ship,  without  reference  to  the  amount 
by  which  the  ship  changes  its  position  from  one  observation  to  another , 
subject  only  to  the  condition,  that  the  distance  between  the  ship  and 
shore  does  not  exceed  the  limits  of  distinct  vision  of  the  signal  to  be 
sighted  from  each  station.  Thus,  a  steamship  may  steam  round  a  circle, 
or  by  other  convenient  evolutions  obtain  the  desired  Headings  (see 
figure);  a  sailing-ship  may  make  a  succession  of  tacks  for  the  same 
purpose;  or  either,  while  riding  at  anchor,  may  swing  to  the  wind  and 
tide,  and  whenever  expedient  be  pulled  round  either  by  warps  or  by  a 
steam-tug. 

It  is  not  necessary  to  restrict  the  observations  on  shore  to  the  use  of 
a  single  station;  it  being  entirely  admissible  to  change  the  instrument 
from  one  position  to  another,  subject  to  the  condition  above  mentioned, 
whenever  it  may  be  expedient,  in  order  to  avoid  obstructions  in  the  line 
of  sight,  such  as  the  masts,  smoke-funnels,  etc.,  on  board. 

88.  Two  Distinct  Cases  of  Reciprocal  Bearings.— 

There  will  be  two  distinct  Cases,  according  to  the  nature  of  the  instru- 
ment used  in  the  shore- observations.    This  may  be— 

1.  An  Azimuth- Compass  similar  to  the  Standard  Compass  on  board,  or 
a  Sur\7eyor's  Compass  adapted  to  observations  on  land,  with  either  of 
which  Magnetic  Bearings  of  the  station  on  board  may  be  taken,  free 
from  Deviation,  but  affected  by  Variation ;  or, 

2.  A  Theodolite  or  Dumb -Compass,  with  which  True  Bearings  may  be 
taken,  free  from  both  Deviation  and  Variation. 

In  the  First  Case,  the  difference  between  any  two  Simultaneous  Bear- 
ing^ is  the  Deviation  of  the  compass  on  board. 

In  the  Second  Case,  there  are  two  modifications,  according  as  the  Bear- 
ings or  Angles  are  measured  from  the  True  Meridian,  or  from  any  Arbi- 
trary Line,  passing  through  the  station.  When  measured  from  the  True 
Meridian,  the  difference  between  any  pair  of  Simultaneous  Bearings  is 
the  Total  Error  of  the  compass  on  board.  When  measured  from  any 
Arbitrary  Line,  the  Differences  of  Simultaneous  Bearings  are  neither 
Total  Errors  nor  Deviations,  but  quantities  which  may  be  reduced 
either  to  Deviations  or  to  Total  Errors,  as  may  be  desired. 

89.  Essential  Requisites  of  the  jflethod  under  the 
First  Case.— The  ordinary  procedure  is  to  use  an  Azimuth-Compass 
for  the  shore-observations,  which  is  furnished  with  a  strong  tripod  fitted 
with  a  fork  to  receive  the  gimbal-ring  of  the  compass.    An  air-compass 
is  not  only  sufficient,  but  rather  more  portable  than  a  liquid  compass  for 
this  purpose.    But,  although  less  nautical,  a  good  surveyor's  compass 
is  preferable  to  either  for  the  shore-observations,  as  is  readily  seen  in 
practice,  and  it  is  much  less  expensive  as  an  extra  instrument. 


76  FINDING    THE    COMPASS-ERROR. 


There  are  two  requisites  which  must  receive  strict  attention : 

First,  in  selecting  stations  on  shore  which  are  known  or  ascertained 
to  be  entirely  free  from  the  influence  of  local  magnetism,  as  due  to 
masses  of  iron  or  other  magnetic  bodies,  whether  visible  or  concealed 
from  view. 

Secondly,  in  comparing  the  shore  compass  with  the  standard  used  on 
board,  so  as  to  determine  what  differences,  if  any,  exist  between  the 
readings  of  the  two  compasses  on  the  same  bearing. 

These  necessary  preliminary  tests  and  comparisons  being  carefully 
made,  the  shore-compass  will  be  ready  for  use. 

i»O,  Essential  Requisites  of  the  Method  under  the 
Second  Case. — With  this  method,  unlike  the  preceding,  there  is  no 
question  of  local  disturbance.  Moreover,  from  the  nature  of  the  instru- 
ment, whether  Theodolite  or  Dumb-Compass,  it  may  be  relied  on  to  the 
nearest  tenth  of  a  degree  when  in  proper  adjustment;  so  that  there  is 
no  comparison  required.  . 

The  stations  may  be -selected  at  points  on  shore  with  reference  solely 
to  their  most  convenient  occupation,  whether  upon  the  docks  or  else- 
where. 

As  already  mentioned,  the  bearings  of  the  compass-station  on  board 
may  be  measured  from  the  True  Meridian,  or  from  any  Arbitrary  Line, 
passing  through  the  station  of  the  instrument. 

a)  The  Angles  Measured  from  the  True  Jfleridiaii.— 

In  this  case  the  graduated  limb  of  the  instrument  is  clamped  with 
its  zero  upon  the  True  Meridian.  Hence,  the  True  Meridian  must  be 
known  in  advance,  and  the  instrument  set  upon  it  whenever  required; 
or  it  must  be  found,  for  a  given  occasion,  at  the  station  of  the  instru- 
ment. Either  is  entirely  practicable.  At  navy-yards,  as  well  as  in 
harbors  and  roadsteads  generally,  convenient  Meridian-Lines  may  be 
established  for  this  purpose,  with  permanent  marks  for  reference,  with 
suitable  station-centres  permanently  marked  upon  the  flagging  or  upon 
stones  set  in  the  ground;  or,  on  any  given  occasion,  the  Meridian  may 
be  found  with  the  requisite  precision  by  means  of  a  computed  Solar 
Azimuth  for  a  particular  instant  of  time,  and  then  laid  down  with  the 
instrument  to  be  used  in  the  observations. 

i>)  The  Angles  Measured  from  any  Arbitrary  Line.— 

For  this  it  is  only  necessary  to  clamp  the  graduated  limb,  without 
regard  to  the  zero,1  and  during  the  observations  to  direct  the  line  of 
sight  to  the  compass-station  on  board,  when  the  readings  obtained  will 
be  True  Angles  from  the  Arbitrary  Line  supposed  to  pass  through  the 
zero  of  the  graduation.  Moreover,  it  will  be  sufficient,  whenever  the 
station  on  shore  is  changed  during  the  progress  of  the  observations, 
to  keep  the  limb  clamped  and  continue  the  Angles  as  before.  But,  since 

1  It  will  be  convenient,  though  not  essential,  to  clamp  so  that  the  zero  shall  fall  out- 
side of  the  field  in  which  the  angles  are  to  be  observed. 


METHOD  BY  RECIPROCAL  BEARINGS.  77 


the  bearings  on  shore  are  taken  without  regard  to  any  known  line  of 
reference,  while  the  position  of  the  ship  and  that  of  the  compass-station 
on  board  are  continually  changing,  it  will  be  necessary  to  reduce  the 
observations  on  board,  in  the  manner  to  be  explained,  to  what  they 
would  have  been  if  they  had  been  made  at  a  fixed  station  and  in  an 
unchangeable  direction. 

91.  Preparations  lor  Observations  by  this  Method. — 

All  observations  by  this  method  must  be  directed  from  the  ship ;  and, 
preparatory  to  undertaking  them,  proper  signals  should  be  arranged  for 
display  on  board  to  guide  the  observer  on  shore.  The  following  signals 
should,  at  the  least,  be  provided  for : 

1.  Prepare  to  observe! 

2.  Observe  and  register ! 

3.  Observation  is  satisfactory ! 

4.  Prepare  to  repeat  the  observation  ! 

5.  Pack  the  instrument  and  return  on  board ! 

Both  observers  should  be  prepared  to  note  the  Time  of  each  observa- 
tion, as  a  check  against  mistakes  of  comparison ;  their  watches  being- 
compared  before  and  after  the  set  of  observations.  It  is  essential  that 
the  two  observations  of  each  pair  be  made  simultaneously.  The  instru- 
ment itself  at  each  station  may  be  sighted  as  the  object  to  be  observed  ; 
but  it  is  better  to  erect  a  more  conspicuous -and  more  nicely  denned 
signal  directly  above  each  instrument.1 

92.  Process    of    Conducting*    Reciprocal    Observa- 
tions.— All  things  being  ready,  and  the  observers  at  their  respective 
stations,  bring  the  ship  carefully  upon  the  desired  Heading,  and,  while 
being  duly  stopped  and  steadied,  make  the  first  signal ;   then,  when 
ready  to  observe,  make  the  second  signal,  upon  which,  at  the  instant, 
the  observer  at  each  instrument,  on  board  and  on  shore,  will  note  his 
Bearing  of  the  other,  and  the  Time  by  watch.    Also,  at  the  same  mo- 
ment, note  the  Ship's  Head  by  the  Standard  Compass,  and  by  each  of  the 
Steering-Compasses.  Note,  also,  the  Angle  of  Heel.  All  being  completed, 
and  the  shore-observation,  as  posted  up,  apparently  satisfactory,  make 
the  third  signal. 

Proceed,  by  bringing  the  ship  upon  a  different  Heading,  making  and 
recording  the  observations  in  a  similar  manner;  and  so  on,  round  the 
compass-circle,  or  until  the  requisite  observations  shall  have  been  made 
for  a  sufficient  number  of  different  Headings. 

As  in  other  cases,  observations  for  Headings  on  the  regular  equidis- 
tant points  by  compass  are  preferable,  but  they  are  not  essential. 

1  It  is  advisable,  as  a  convenient  precaution,  for  the  observer  on  shore  to  chalk  each 
observation  upon  a  blackboard,  so  that  it  may  be  read  (using  a  glass  if  necessary)  by 
the  observer  on  board  ;  by  which  means,  if  there  should  be  any  apparent  inconsistency, 
the  observation  may  at  once  be  repeated,  and  the  necessity  thereby  avoided  for  again 
swinging  the  ship. 


78  FINDING   THE   COMPASS-ERROR. 


93.  RecluctioBi  of  Reciprocal  Observations.— There  are 
three  Cases  to  be  considered,  according  to  the  nature  of  the  shore-obser- 
vations. 

a)  Magnetic  Observations  on  Shore. — In  this  Case  the  Difference  between 
the  corresponding  Ship  and  Shore  Bearings  is  the  Compass-Deviation, 
which  will  be  marked  E  or  W9  according  as  the  observation  on  board 
falls  to  the  Left  or  Right  of  that  on  shore. 

b)  Angles  Measured  from  the  True  Meridian  on  Shore. — Here  the  Differ- 
ence between   Simultaneous  Bearings  is  the  Compass-Error,  which  is 
marked  E  or  W,  according  as  the  observation  on  board  falls  to  the  Left 
or  Eight  of  that  on  shore. 

c)  Angles  Measured  from  any  Arbitrary  Line  on  Shore.— -First,  note  the 
least1  among  the  Shore  Angles,  and  regard  it  as  the  bearing  of  the  Zero- 
Line,  or  line  of  reference;  then,  compare  with  this  all  the  other  Angles, 
setting  down  the  corresponding  Differences  in  a  column  of  Reduction* 
for  Shore  Instrument,  and  marking  them  R  or  L,  according  as  the  Zero- 
Line  falls  to  the  Right  or  Left  of  the  Angles  compared  with  it. 

Next,  apply  these  reductions  to  the  corresponding  Bearings  from  the 
compass  on  board — to  the  Right  or  Left,  according  as  they  are  marked  R 
or  L ;  which  call  the  Reduced  Bearings  by  the  Standard  Compass.  There 
will  now  be  twro  Cases,  according  as  the  observations  are  made  on  Equi- 
distant Headings  or  at  Irregular  Intervals. 

First :  Observations  on  Equidistant  Headings. — Take  the  Mean  of  the 
Reduced  Bearings,  whether  32,  16,  or  8,  and  this  will  be  the  Magnetic 
Bearing  of  the  station  on  shore;  then,  the  Differences  between  this  and 
the  other  actual  Reduced  Bearings  will  be  the  Deviations,  to  be  marked 
E  or  TT,  according  as  the  latter  Bearings  fall  to  the  Left  or  Eight  of  the 
Mean  Bearing. 

Secondly :  Observations  at  Irregular  Intervals. — Find  the  Reduced  Bear- 
iugs  by  the  Standard  Compass,  as  in  the  previous  case;  thence,  proceed 
as  follows: 

Compare  the  Reduced  Bearings  with  that  of  the  zero-line,  marking  the 
Differenced  or  TF,  according  as  the  former  fall  to  the  Left  or  Right  of  the 
zero-line. 

With  these  Differences  as  Ordinates,  and  the  Headings  as  Distances, 
construct  a  curve,  in  the  manner  to  be  explained  hereafter,  from  which 
deduce  the  Mean  Difference  on  a  series  of  Equidistant  Headings;  set 
off  this  Difference  according  to  its  name,  and  draw.a  parallel  to  the 
Line  of  Distances;  then,  the  Ordinates  to  the  curve  with  reference  to 
this  new  line  of  distances,  for  whatever  Headings  taken  off,  will  be  the 
Compass-Deviations  corresponding  to  those  Headings. 


1  Any  other  Shore-Angle  may  be  chosen  as  that  of  the  zero-line,  as  may  be  deemed 
most  convenient.  The  least  makes  the  Differences  or  Reductions  take  the  same  name; 
all  R  orallL. 


METHOD  BY  RECIPROCAL  BEARINGS. 


94.  Examples  of  the  JTIethocl  by  Reciprocal  Bear- 
ings. 

Ex.  1.— 1868,  June  18 :  At  Erie,  on  Lake  Erie,  in  Lat.  42°  N,  Long.  80°  W, 
U.  S.  Ship  Michigan  (iron-built  paddle-wheel  steamer).  Swung  for  a  set  of  Compass- 
Deviations,  by  Reciprocal  Bearings. 

Observations.  Reductions. 


Time. 

by  Standard 
Compass. 

Simultaneous  Bearings. 

Standard 
Compass  on 
Board. 

Azimuth* 
Compass  on 
Shore. 

h    m 

o 

o 

II    13 

North. 

S  60.3  E 

N  59.0  W 

15 

NbyE 

56.7 

59-° 

27 

.NNE 

59-o 

33 

NE  by  N 

48.7 

59-° 

12      8 

NE 

42.8 

59-5 

ii 

NE  by  E 

42.0 

59-5 

*4 

ENE 

41.5 

60.0 

17 

E  by  N 

38.0 

60.0 

12    19 

East. 

S  35-3  E 

N  60.5 

22 

E  by  S 

37-2 

60.5 

25 

ES'E 

36.5 

61.0 

33 

SE  by  E 

37-2 

61.0 

•    45 

SE 

40.7 

60.5 

51 

SE  by  S 

40-5 

60.0 

54 

SSE 

49-3 

60.0 

SbyE 

53-7 

60.0 

I       2 

South. 

S'57-3  E 

N  60.0  W 

6 

S  by  W 

64-3 

60.0 

10 

SSW 

70.0 

60.0 

14 

SW  by  S 

76.0 

60.0 

\-[ 

SW 

79-7 

59-5 

22 

SW  by-W 

83.0    ' 

59-o 

27 

ws'w 

85.0 

58.5 

36 

W  by  S 

86.7 

58.0 

i  39 

West. 

S  86.0  E 

N  57-5  W 

51 

WbyN 

857 

57-o 

54 

WNW 

83.0 

57-0 

2    4 

NW  by  W 

82.7 

56.5 

7 

NW 

78.0 

56.5 

ii 

NW  by  N 

75-3 

15 

NNW 

70.0 

57-5 

19 

N  by  W 

67.8 

58.0 

Deviation  of 

the  Standard 

Compass. 


1.3  E 
2.3  W 

6.0 
10.3 
16.7 

ll'.l 

22.0   W 


25.2  W 

23-3 
24-5 

23.8 

19.8 

19-5 
10.7 

6.3  W 

2.7  W 

4-3  E 
10.0 
16.0 

20.2 
24.0 
26.5 

28.7  E 

28.5  E 

28.7 

26.0 

26.2 

21.5 

18.- 

12.5 

9.8  E 


Explanations. — The  shore-instrument  in  this  case  being  an  Azimuth-Compass,  the 
Bearings  are  Magnetic,  and  the  Differences  of  Simultaneous  Bearings  are  Deviations  of 
the  Standard  Compass  on  board. 

There  are  certain  anomalies  in  the  results,  which  become  more  obvious  by  a  Graphical 
Construction,  and  at  the  same  time  admit  of  being  corrected  and  the  results  reduced  to 
mf>re  probable  values. 

The  Standard  Compass  was  placed  on  top  of  pilot-house  on  hurricane-deck,  28  feet 
above  spar-deck. 


80 


FINDING    THE    COMPASS-ERROR. 


JEx.  2. — 1868,  June  18:  At  Erie,  on  Lake  Erie,  in  Lat.  42°  N,  Long.  80°  W, 
U.  S.  Ship  Michigan :  The  Azimuth-Compass  used  on  shore  in  Ex.  i  supposed  to  have 
been  replaced  by  a  Dumb-Compass,  and  the  Angles  measured  from  an  Arbitrary  Lino. 


Observations. 


Reductions. 


Ship's  Head 
by  standard 
Compass. 

Simultaneous  Bearings. 

Standard 
Compass 
on  Board. 

Dumb-Com- 
pass on 
Shore. 

o 

o 

North. 

S  60.3  E 

39-o 

NbyE 

56.7 

39-o 

NNE 

53-° 

39-o 

NEbyN 

39-o 

NE 

42.1 

39-5 

NEby  E 

42.0 

39-5 

'  ENE 

4I-5 

40.0 

EbyN 

38.0 

40.0 

East. 

S  35-3  E 

40-5 

E  by  S 

37-2 

40-5 

ESE 

36.5 

41.0 

SE  by  E 

37-2 

41.0 

SE 

40.7 

40-5 

SEby  S 

40-5 

40.0 

SSE 

49-3 

40.0 

S  by  E 

53-7 

40.0 

South. 

S  57-3  E 

40.0 

S  by  W 

64-3 

40.0 

ssw 

70.0 

40.0 

SVV  bv  S 

76.0 

40.0 

SW 

79-7 

39-5 

SW  by  W 

83.0 

39-o 

WSW 

85.0 

38.5 

Wby  S 

86.7 

38.0 

West. 

S  86.0  E 

37-5 

Wby  N 

857 

37-° 

WNW 

83.0 

37-° 

NW  by  W 

82.7 

36.5 

NW 

78.0 

•36* 

NW  by  N 

75-3 

37-o 

NNW 

70.0 

37-5 

Nby  W 

67.8 

38.0 

Reductions 
to  Zero-Line 
of  Shore-  An- 
gles. 

Reduced 
Bearings 
by  Standard 
Compass. 

Deviations 
of  the  stand- 
ard Compass. 

o 

2.5  R 

S  57-8  E 

o 

0.4  W 

2-5 

54-2 

4.0 

2-5 

5°-5 

7-7 

2-5 

46.2 

I2.O 

3-o 

39-8 

I8.4 

3-° 

39-o 

19.2 

3-5 

38.0 

20.2 

3-5 

34-5 

23-7  w 

4.0  R 

S  31.3  E 

26.9  W 

4.0 

33-2 

25.0 

4-5 

32.0 

26.2 

4-5 

32-7 

25-5 

4.0 

36.7 

21.5 

3-5 

37-o 

21.2 

3-5 

45-8 

I2.4 

3-5 

50.2 

.     8.0  W 

3-5  R 

S  53-8  E 

4-4  W 

3-5 

60.8 

2.6  E 

3-5 

66.5 

8-3 

•    3-5 

72.5 

14-3 

3-° 

76.7 

18.5 

2-5 

80.5 

22.3 

2.O 

83.0 

24-8 

i-5 

85.2 

27.0  E 

1.0  R 

S  85.0  E 

26.8  E 

0-5 

85.2 

27.0 

°-5 

82.5 

24-3 

0.0 

82.7 

24-5 

o.o 

78.0 

19.8 

0-5 

74.8 

16.6 

l.O 

69.0 

10.8 

1.5 

66.3 

8.1  E 

Mean  S  58°.2  E 

Explanations.— The  Least  Angle  of  the  shore-observations  is  *36°.5,  which  is  taken 
as  that  of  the  Zero-Line.  Comparing  this  with  all  the  other  Shore-Angles,  we  get  the 
Reductions  in  Col.  I.  These  are  all  marked  R,  because  the  direction  of  the  Zero-Line 
falls  to  the  Right  of  all  the  other  Angles. 

Next,  these  Reductions  are  applied  to  the  Right  of  the  corresponding  Bearings  by  the 
Standard  Compass,  and  we  get  Col.  II 5  the  Mean  of  which,  or  S  58°.2  E,  is  the  Magnetic 
Bearing  of  the  shore-station. 

Finally,  comparing  the  Reduced  Bearings  of  Col.  II  with  the  Mean,  we  get  the  Devia- 
tions in  Col.  III. 


METHOD  BY  RECIPROCAL  BEARINGS, 


81 


Ex.    3.—  1868,  June  18  :    At  Erie,  on  Lake  Erie,   in  Lat.  42°  N,   Long.  80°  W, 

U.  S.  Ship  Michigan  :  The  ship  swung  upon  Irregular  Headings,  and  the  Angles  on  shore 

wtill  measured  from  an  Arbitrary  Line  with  a  Theodolite. 

Observations.                                              Reductions* 

Simultaneous  Bearings. 

Ship's  Head 
by  standard 
Compass. 

Reductions 
to  Zero-Line 
of  Shore-An- 
gles. 

Reduced 
Bearings 
by  Standard 
Compass. 

Differences  of 
the  Standard 
Compass. 

Standard 
Compass  on 
Board. 

Theodolite 
on  Shore. 

o 

o 

o 

o 

c 

O 

N  10  E 

S  57-3  E 

43-2 

1.7  R 

S  55.6  E 

14.1  E 

, 

22 

53-5 

44-8 

3-3 

50.2 

8.7 

40 

45-7 

42.8 

444 

2.9  E 

53 

42.4 

43-9 

2.4      - 

40.0 

1.5  W 

41.2 

46.0 

4-5 

36.7 

4.8 

88 

36.9 

45-3 

3-8 

33-  i 

8.4 

S  73  E 

S  36.1  E 

43-9 

2.4 

33-7 

7-8 

56 

37-0 

44.0 

2-5 

34-5 

7.0 

40 

4°-3 

42.6 

i.i 

39-2 

2.3  W 

27 

47-2 

45  -1 

3-6 

43-6 

2.1    E 

10 

54-7 

44-8 

3-3 

9-9 

S    5  W 

S  61.0  E 

42.8 

1-3 

59-7 

18.2  E 

22 

69.8 

42.8 

'•3 

68.5 

27.0 

36 

75-3 

42.5 

I.O 

74-3 

32-8 

54 

82.1 

43-9 

2.4 

79-7 

38.2 

65 

84.5 

43-  ! 

1.6 

82.9 

41.4 

78 

86.3 

^42.2 

0.7 

85.6 

44.1 

89 

85.9 

0.0 

85.9 

444 

N  75  W 

S  85.7  E 

41.7 

0.2 

85.5 

44.0  E 

62 

82.8 

41.7 

0.2 

82.6 

41.1 

50 

80.2 

42-3 

0.8 

794 

37-9 

36 

76.6 

42.4 

0.9 

75-7 

34-2 

18 

68.5 

42.1 

0.6 

67.9 

26.4 

4 

65.1 

45.8 

4.3  R 

60.8 

19.3  E 

Explanations.—  Having  found  the  Reduced  Bearings  of  Col.  II,  as  in  the  preceding 

Example,  we  next  Compare  these  with  the  Angle  of  the  Zero-Line  (*4i°.s)  and  get  the 

Differences  in  Col.  III. 

11 


82  FINDING  THE  COMPASS-ERROR. 


95.  Dependence  to  be  placed  on  Results.— With  the  use 
of  an  Azimuth-Compass  at  the  shore- station,  there  is  always  consider- 
able uncertainty  relative  to  the  Magnetic  Meridian,  from  which  the  Bear- 
ings  are  supposed  to  be  measured,  unless  the  requisite  means  are 
employed  to  have  a  strict  comparison  of  the  two  compasses  made  on 
shore.    Not  only  is  it  necessary  to  detect  the  possible  influence  of 
Local  Magnetism,  but  to  note  the  actual  differences  of  the  shore  Com- 
pass- -Readings  on   different  parts  of  the  compass-card,  if  we  would 
be  at  all  sure  of  our  results.     With  these  comparisons  made,  and  the 
ascertained  Errors,  if  any,  properly  applied  as  corrections,  the  shore- 
observations  should  be  depended  on. 

With  the  use  of  a  Theodolite  or  Dumb-Compass,  on  the  other  hand, 
a  more  satisfactory  precision  may  be  attained.  It  is  not  difficult  for  the 
Navigator,  with  a  reliable  Geographical  Position  and  a  good  Local  Time, 
to  determine  his  True  Meridian  on  shore  to  the  nearest  minute  5  and 
this  may  be  established  by  Meridian-Marks,  and  worked  from  to  a  cor- 
responding precision,  with  a  Theodolite  reading  to  minutes  in  proper, 
adjustment.  Even  with  a  Dumb-Compass,  reading  only  to  tenths  of 
degrees,  the  Bearings  should  be  depended  on  to  the  nearest  tenth  of  a 
degree. 

There  will  not  be  the  same  degree  of  precision,  if,  instead  of  working 
from  the  True  Meridian,  the  Shore- Angles  are  measured  from  an  arbi- 
trary line.  In  this  case  a  certain  constant  Error  will  be  introduced  and 
merged  in  the  Errors  of  Compass  Adjustment  and  Observation,  in 
consequence  of  the  necessity  of  employing  a  Mean  Bearing  or  Difference 
as  deduced  from  the  results  of  the  observations.  The  Error  from  this 
source  will  be  less  or  greater  according  to  the  greater  or  less  care  and 
nicety  observed  in  conducting  the  operations.  In  general,  it  will  rarely 
be  so  much  as  a  degree,  including  the  Errors  of  Adjustment  and  Obser- 
vation 5  and,  since  it  affects  all  the  results  alike,  it  may  commonly  be 
disregarded. 

96.  To  find  the  True  Ifleridian. — There  are  various  modes 
of  procedure  for  finding  the  True  Meridian.    It  will,  however,  be  quite 
sufficient,  for  the  present  purpose,  to  make  use  of  means  to  which  the 
Navigator  is  well  accustomed,  namely :   That  of  obtaining  the  True 
Azimuth  of  the  Sun,  or  other  celestial  object,  at  a  particular  moment  of 
Local  Time  ;  thep,  of  setting  off  the  Azimuth  at  the  designated  mo- 
ment, upon  a  Theodolite  or  Dumb  Compass.    The  whole  process  may 
be  conducted  as  follows  : 

a)  Compute  in  advance,  for  a  designated  Local  Time,  the  True  Azi- 
muth of  the  Sun  or  other  celestial  body  (19);  and  add  or  subtract  the 
Semi-Diameter,  according  as  the  Azimuth  is  greater  or  less  than  90°.    The 
resulting  Azimuth  will  be  that  of  the  inner1  limb,  if  the  body  have  a 
disk  (Sun  or  Moon). 

b)  Having  selected  a  convenient  observing-station,  set  the  Theodolite 
or  Dumb-Compass  at  the  chosen  point;  then  turn  off'  the  ascertained 

1  That  towards  the  Meridian. 


METHOD    BY    RECIPROCAL    BEARINGS.  83 


True  Azimuth  from  the  line  of  zeros,  and  carefully  clamp  the  alidade 
(telescope  or  sight-  vanes  as  the  case  may  be),  and  level  the  instrument 
for  observation. 

c)  As  the  designated  Time  approaches  for  which  the  True  Az.  is  found, 
point  the  line  of  sight  a  few  minutes  in  advance  to  the  inner  limb  of 
the  celestial  body,  turning  the  whole  instrument  upon  its  spindle,  and 
keeping  upon  the  observed  point  of  contact,  by  following  the  motion  of 
the  body,  till  the  precise  moment  of  time  is  reached  ;  then  make  fast 
the  lower  or  spindle  clamp. 

d)  Uuclamp  the  line  of  sight,  turn  to  zero,  and  again  clamp;  it  will 
now  be  in  the  True  Meridian.    Set  up  a  Meridian-Mark,  in  the  direction 
of  the  line  of  sight,  fifty  to  a  hundred  yards  in  advance.     Uuclamp  and 
turn1  the  line  of  sight  180°;  uuclamp,  and  then  set  up  another  Meridian- 
Mark   in  the  opposite  direction.     Finally,  mark  the  station  in  a  perma- 
nent manner  on  the  ground  below  the  centre  of  the  instrument,  and  it 
will  be  ready  for  use. 

97.  Remark:   To  set  the  Instrument  in  a  True  Me- 
ridian. already  established.—  If  the  True  Meridian  be  already 
established,   proceed    as    follows   to   place   the   instrument:    Set   the 
instrument  over  the  station  (d)  with  a  centre-plumb,  and  level  it;  then 
clamp  the  alidade  to  zero,  and  direct  the  line  of  sight  to  one  of  the 
permanent  Meridian-Marks  (d),  turning  the  whole  instrument  upon  the 
spindle,  and,  when  carefully  bisected,  make  fast  the  lower  or  spindle 
clamp.     The  instrument  will  be  ready  for  use. 

98.  Example  of  Finding  the  True    Meridian  of  a 
Shore-Station. 

Ex.  1.  —  1872,  February  20:  Navy-  Yard  Dock,  Washington,  iii  Lat.  38°  52^4  N,  Loii£ 
y6059'.6  W.  It  is  required  to  find  the  True  Meridian  from  a  True  Azimuth  of  the  Sun 
at  4b.o  P.  M.  Loc.  A.  T.  :  Watch  fast  on  LOG.  M.  T.  im  14s. 

Loc.  A.  T.        4h  om      ©'sDec.          S  11°    2^9  {—54"  Eq.  of  T.        -f-  14™  i8.;]—  os.2; 
Long.  +58         Rednforg1'.!  8.2    486    Rednfor9h.i  2.5        2.43 

Gr.  date      20  9    8         Red.  Dec.  10    54.7         5     Red.  Eq.  T.         13   59  .2  3 

or,  9ll.l         Pol.  Dist.  100    54.7  -f-  to  A.  T. 

Eq.  of  T.          +  oh  13™  59s 

Loc.  M.  T.  4    13     59     Pol.  Dist.  100°  54'.  7 

Er.  of  W.  fast      +     i     14     Co-Lat.        51     7.6 
T.   byW.  4    15     13      Diff.  49  47.1 

y2  Diff.        24  53.5     sili      9.6242    cos  9.9577 

]/z  Sum        76     i  .1      cosec  0.0131     sec  0.6168 

Yz  H.  A.         21'  om        cot      0.2386    cot  0.2386 

tan     9.8759    tan  0.8131 

TrueAz.       Nii8°io/W  X      36°  55'      Y  81°  15' 

0's  Semi-D.        +    16 
True  Az.       N  118  26  W  Inner  Limb 
or,          S    61  34  W 

A  few  minutes  before  4h  15™  i38  by  Watch,  place  the  theodolite  firmly 
in  position  at  the  station  chosen  for  the  shore-observations,  and  level  it. 


is  unnecessary  with  a  theodolite  whose  telescope  may  be  revolved  —  transit- 
like. 


84  FINDING   THE    COMPASS-ERROR, 


Next,  turn  off  the  Angle  61°  34'  from  o,  and  clamp  the  limb,  leaving 
the  lower  or  spindle  damp  free. 

.Now,  turning  the  whole  instrument  upon  the  spindle,  pointing  to  the 
Sun,  and  bringing  the  vertical  wire  of  the  telescope  into  contact  with 
its  inner  limb,  follow  its  motion,  carefully  preserving  contact,  until  the 
moment  4h  15™  43e  arrives,  as  shown  by  watch;  then  make  fast  the  lower 
clamp. 

Unclamp  the  limb,  and  turn  the  telescope  back  to  o,  when  it  will  be 
in  the  True  Meridian.  Set  up  a  Meridian -Mark  in  the  direction  of  the 
line  of  sight  and  mark  the  centre  of  the  instrument,  as  shown  by  the 
plumb,  on  the  ground  below.1 

The  instrument  is  ready  for  the  shore-observations,  the  lower  clamp 
remaining  all  the  time  fixed. 

Remark. — We  might  proceed  in  the  direct  manner,  by  first  making 
the  observation  to  a  noted  time  by  watch,  subsequently  computing  the 
True  Azimuth,  and,  finally,  turning  it  off  on  the  theodolite.  The  only 
objection  to  this  mode  of  procedure,  is  the  necessity  of  allowing  the 
instrument  to  stand  and  the  danger  of  its  disturbance  while  making  the 
computation. 

D.  RELATIVE  ADVANTAGES  OP  THE  SEVEKAL  METHODS  OF  FINDING  SERIAL  COMPASS- 
ERRORS.  '    * 

We  shall  briefly  sum  up  the  relative  advantages  of  the  principal 
Methods  for  finding  Serial  Compass-Errors. 

99.  Jlletliocls  by  Celestial  Azimuths.— The  observations 
by  these  methods,  and  preferably  by  Time-Azimuths,  are  certainly  the 
most  convenient.  They  are  all  made  on  board  ship.  The  ship  may  be 
at  her  anchorage,  or  standing  off  outside,  or  at  sea ;  and  the  different 
Headings,  on  which  the  observations  are  made,  may  be  had,  either  by 
swinging  about  her  anchor,  by  steaming  round,  or  by  tacking  under  sail. 

When  near  the  land,  these  methods  are  susceptible  of  considerable 
accuracy.  The  True  Azimuths  may  be  depended  on  within  i  3'  round 
the  entire  compass-circle,  the  Data  being  reliably  known  and  the  condi- 
tions of  favorable  results  being  duly  satisfied  (60).  The  Compass- Azi- 
muths are'taken  under  the  most  .favorable  circumstances  for  precision 
of  results. 

The  only  drawback  to  this  method,  in  comparison  with  the  others, 
while  superior  in  every  particular  besides,  is  the  supposed  labor  of  ob- 
taining the  series  of  True  Azimuths.  And  yet,  a  set  of  thirty-two  Azi- 
muths may  be  had,  with  the  aid  of  the  Tables  here  given,  in  a  half- 
hour  or  less.2 

1  If  a  dumb-compass  instead  of  a  theodolite  bad  been  used,  we  should  have  turned 
off  instead  6i°.6  and  clamped  the  alidade,  leaving  the  spindle  free  ;  otherwise,  the  de- 
scription of  the  text  is  equalty  applicable  to  this  instrument,  only  substituting  "  alidade  " 
for  "limb,"  and  "sight- vanes"  for  "telescope." 

2  Navigators  not  (infrequently  devote  the  befit  part  of  a  day,  sometimes  two  of  them, 
at  a  "  station  "  to  the  laborious  and  disagreeable  duty  of  "  swinging  ship  "  for  observa- 
tions of  a  distant  object,  without  always  obtaining  results  which  they  can  regard  as 


RELATIVE  ADVANTAGES  FOR  SERIAL  COMPASS-ERRORS.       85 


At  sea,  of  course  none  but  Celestial  Azimuths  are  available  for  Com- 
pass-Error ;  and  the  small  labor  of  getting  a  set  wholly  or  partially 
round  the  compass  by  the  Method  of  Time-Azimuths,  should  certainly 
leave  no  Navigator  of  a  ship,  whenever  overtaken  by  prolonged  thick 
weather,  without  recent  determinations  of  the  Errors  on  the  Compass- 
Courses  likely  to  be  sailed  at  such  a  time. 

0 OO.  method  by  Direct  Bearings. — The  Method  by  Direct 
Bearings  has.  also  the  convenience  of  requiring  no  observer  on  shore, 
but  it  has  the  disadvantage  of  requiring  a  particular  station,   com. 
inonly  at  a  considerable  distance  from  the  usual  anchorage,  and  of  being 
attended  by  a  large  expenditure  of  time,  labor,  and  patience  in  warp- 
ing the  ship  about,  especially  if  a  large  one.     It  has,  moreover,  the  dis- 
advantage of  being  liable  not  only  to  the  Constant  Error,  whatever 
that  may  be,  in  getting  the  True  Bearing  of  the  object,  but  also  to  in- 
accuracies in  the  Compass-Bearings,  from  the  more  serious  Errors  of 
Parallax ;  errors  which  are  always  variable  and  occasionally  uncertain, 
and  which  are  frequently  much  too  large  to  be  admitted,  with  any  re- 
gard to  precision  of  results.1 

101.  Ilethod  by  Reciprocal  Bearings.— The  use  of  this 
method  is  attended  with  the  disadvantage  of  requiring  observers  on  shore 
in  addition  to  those  on  board.    But  it  admits  of  being  employed  at  the 
ordinary  anchorage,  either  by  swinging  at  an  anchor,  or  by  steaming 
about  when  the  water-way  permits.    Neither  Parallax  nor  any  other  con- 
sideration need  interfere  with  entire  freedom  in  the  movements  of  the 
ship,  during  the  operations  of  placing  her  upon  different  Headings,  ex- 
cept that  of  keeping  the  distance  between  the  ship  and  station  on  shore 
within  the  limits  of  distinct  vision. 

With  the  use  of  an  instrument  for  measuring  the  Bearings,  either 
from  the  True  Meridian  or  from  any  Arbitrary  Line  at  the  station  on 
shore,  not  only  is  the  choice  of  that  station  independent  of  all  re- 
gard to  the  influence  of  Local  Magnetism,  but  the  Angles  admit  of 

"  entirely  satisfactory  ; "  which  in  fact,  are  quite  too  frequently  utterly  unreliable  and 
valueless.  Whereas  by  merely  steaming  about  at  their  anchorage  (not  being  too  near 
auy  other  ship),  they  might  get  a  perfect  set  of  observations  of  the  Sun  in  two  or 
three  hours  after  sunrise  or  before  sunset ;  and  another  half-hour  or  so  would  enable 
them  to  reduce  their  observations  and  obtain  a  Table  of  Compass-Deviations. 

1  See  the  interesting  and  suggestive  description,  by  the  late  Dr.  Scoresby  (pp.  168-182 
of  Journal  of  a  Voyage  to  Australia,  etc.,  London,  1859),  of  his  arduous  labors  for  more 
than  a  week  at  Melbourne,  Australia,  in  determining  Magnetic  Bearings  of  distant 
objects  and  in  swinging  the  noble  iron  ship  Royal  Charter.  With  an  admitted  Par- 
allax of  o°.3  to  o°4  on  some  of  his  Headings;  with  an  apparent  uncertainty  of  perhaps  a 
whole  degree  in  the  Magnetic  Bearings  of  his  objects ;  with  probably  considerable  errors 
i'n  the  Compass-Bearings  from  the  manner  in  which  they  were  taken ;  and  with  an  im- 
mense amount  of  labor  and  anxiety  on  his  part,  aided  by  gangs  of  men,  boats,  kedges, 
and  warps,  he  obtained  a  series,  as  was  to  be  expected,  of  not  very  satisfactory  results. 
And  yet,  with  steam  up  on  one  of  those  days,  and  with  two  or  three  hours  devoted  to 
a  series  of  Solar  Azimuths,  what  superior  results,  and  what  a  saving  of  labor,  worry, 
and  anxiety  might  have  been  realized  ! 


86  FINDING    THE    COMPASS-ERROR. 


being  measured  with  greater  precision.  It  is  quite  possible  to  have  the 
True  Meridian  laid  down  on  shore  within  a  minute  of  arc  5  and  the 
Angles  may  be  measured  with  equal  accuracy,  using  a  theodolite  for 
the  purpose. 

When  the  True  Meridian  has  not  been  already  laid  down,  the  Angles 
may  be  measured  from  Arbitrary  Lines  ;  but,  in  this  case,  the  Mean  of 
the  Reduced  Compass  Bearings  having  to  be  used  as  the  Magnetic  Bear- 
ing of  the  assumed  zero-line,  the  results  may  involve  a  greater  Constant 
Error. 

Otherwise,  in  working  from  the  Magnetic  Meridian  on  shore,  this 
method  is  not  only  more  restricted  in  the  choice  of  a  station,  but  is  lia- 
ble, without  great  care  and  a  troublesome  comparison  on  shore,  to  consid- 
erable Errors  from  Local  Magnetism,  and  differences  in  Compass- .Read- 
ings. 

In  conclusion,  since  one  or  the  other  of  the  two  Methods  by  Celestial 
Azimuths  and  Reciprocal  Bearings  is  sufficient  for  all  probable  circum- 
stances, and  since  either  of  these  Methods  may  be  so  conducted  as  to 
insure  the  requisite  precision  of  results,  there  would  seem  to  be  little  it 
any  occasion  for  resort  to  the  less  accurate  Method  by  Bearings  of  a 
Distant  Object. 


APPENDIX.  87 


APPENDIX. 


COMPASS  COMPARISONS. 

1O3.  Definition.— The  Comparison  of  one  compass  with  another 
is  the  Difference  of  their  Readings,  one  being1  taken  as  Standard  for 
Azimuths  or  Bearings  of  the  same  object:  It  is  marked  E or  W,  according 
as  the  Bearing  by  the  Compared  Compass  falls  to  ib&  Left  or  Right  of 
the  Bearing  by  the  Standard,  the  eye  being  supposed  at  the  centre  of 
the  latter's  compass  card. 

A  distinction  exists  between  the  Comparison  of  Compasses  on  board 
and  on  shore,  which  will  be  separately  considered. 


A.    COMPASS-COMPARISONS   ON    BOARD    SHIP. 

10SI.  Characteristics  and  Uses  of  Comparisons.— From 

what  has  already  been  said,  as  to  the  nature  of  the  Compass-Error,  it 
may  be  inferred  that,  whenever  there  are  several  compasses  set  up  for 
use  in  different  parts  of  the  same  ship,  they  may  generally  be  expected 
to  show  different  Readings  on  the  same  Heading  of  the  ship,  and  each  to 
vary  for  different  Headings.  This  has  been  fully  explained  in  the  pre- 
ceding Chapters. 

A  Compass-Comparison  on  board  serves  several  useful  purposes,  of 
which  the  two  following  may  be  specially  mentioned  : 

First,  in  converting  a  Course  by  Standard  into  an  equivalent  Course 
by  Compared  Compass;  or,  vice  versa. 

Secondly,  in  finding  the  Error  of  a  Compared  Compass,  whenever  we 
have  its  Comparison  with  the  Standard  and  the  Error  of  the  latter. 

In  practice,  the  ship's  Head  is  the  object  observed  in  all  Compass- 
Comparisons  on  board.  For  this  there  are  two  reasons  : 

First,  because  the  Compass-Readings,  and  therefore  the  Differences 
between  them,  depend  on  the  direction  of  the  ship's  Head  ;  and. 

Secondly,  because  the  Lubber-Line,  which  represents  the  ship's  Head 
for  each  compass,  is  the  only  object  which  can  be  either  conveniently  or 
accurately  observed  with  the  Steering  and  other  Compasses,  not  fur- 
nished with  the  sight- vanes  of  an  Azimuth-Circle. 

1O4.  To  make  a  Compass-Comparison. — Accordingly, 
the  only  necessary  provision  for  Compass-Comparisons  on  board  is,  to 
note  the  Headings  of  the  ship  by  the  different  compasses  for  which  com- 
parisons are  desired,  whenever  observations  for  Error  are  made  with 
the  Standard  or  Azimuth  Compass  (10). 

The  Rule  for  the  Comparison  is  comprised  in  the  preceding  Defini- 
tion,. 


88 


COMPASS-COMPARISONS. 


1O5.  Examples  of  Compass-Comparisons. 


-Ex.  1. — Stand.  Compass      N  47°  E 

Steering-Compass  N  41    E 

COMPARISON          6  E 

Ex.  2.— Stand.  Compass      S  67°  E 

Steering-Compass  S  59   E 

COMPARISON          8   W 

Ex.  3.— Stand.  Compass       N  W 

Steering-Compass  N  N  W  %  .W 
COMPARISON         i  %  pts.  W 


Ex.  4.— Stand.  Compass      S  11°  E 

Steering-Compass   S    2    W 

COMPARISON         13    AV 

Ex.  5.— Stand.  Compass      N  89°  W  .; 
Steering-Compass   S  87    W 
COMPARISON          4    E 

Ex.  6.— Stand.  Compass      N  E  by  E 

Steering-Compass  E  N  E  |/j  E 
COMPARISON         i  !••£  pts.  W 


106.  To  convert  a  Given  Course  iUy  Standard  Com- 
pass into  an  Equivalent  Course  by  a  Compared  Com- 
pass, and  Reciprocally.— An  important  application  of  the  Com- 
parison of  Compasses  is  found  in  the  conversion  of  a  Standard-Compass 
Course  into  an  equivalent  Coinpared-Compass  Course.     Thus,  it  com- 
monly happens,  in  practice,  that  the  Sailing-Courses,  while  given  for 
the  Standard  Compass,  must  be  set  for  the  Helmsman  on  the  Steering- 
Compass.    This  conversion  of  the   given  Course  is  made  in  the  fol- 
lowing manner: 

Rule  :  Suppose  the  eye  at  the  centre  of  the  Standard  Compass> 
looking  along  the  line  of  the  given  Course  or  Heading ;  then  apply  all 
East  Comparisons  to  the  left  hand,  and  all  West  Comparisons  to  the  right 
hand. 

For  Reciprocal  Conversions,  apply  the  comparison  to  the  Course  by 
*  Compared  Compass  in  the  contrary  manner,  and  the  result  will  be  the 
equivalent  Course  by  Standard  Compass. 

107.  Examples  of  Direct  and    Reciprocal   €.-011  ver- 
sions. 


Ex.  1.— The  Course  or  Heading  by 
Standard  is  NE,  and  the  Comparison  of 
Steering-Compass  is  6°  E  :  Required,  the 
Course  by  Steering-Compass. 

Course  by  Standard  N  45°  E 

Comparison  of  Sceering-Comp.         6    E 
Course  by  Steering-Compass     N  41     E 


Ex.  2. — The  Course  by  Steering-Com- 
pass is  NNW,  and  its  Comparison  with  the 
Standard  is  11°  W:  Required,  the  Course 
by  Standard. 

Course  by  Steering-Compass  N  22°. 5  W 
Comparison  of  Steering-Comp.  n  .o  W 
Course  by  Standard  N  33  .5  W 


1O8.  Rule:  To  find  the  Error  or  Deviation  of" a  Com- 
pared Compass. — Having  the  Comparison  of  any  compass  with  the 
Standard,  and  the  Error  or  Deviation  of  the  latter,  the  Error  or  Devia- 
tion of  the  Compared  Compass  is  found  as  follows: 

Take  the  Sum  or  Difference  of  the  Comparison  and  Error,  or  Devia- 
tion, according  as  they  have  the  same  or  different  names;  and  the  result 
will  be  the  Error  or  Deviation  of  the  Compared  Compass,  which  mark 
with  the  name  of  the  greater  number. 


COMPASS-COMPARISONS. 


89 


1O9,  Examples  of  Finding  the  Errors  of  Compared 


Compasses. 

Ex.  1. — A  Comparison  of  Steering-Com- 
PHSS  is  3/  pt.  W  with  the  Standard,  and 
the  Error  of  the  latter  i  %  pts.  W :  Re- 
quired, the  Error  of  the  Steering-Compass. 
Comparison  of  Steer.  Comp.    %  pt.  W 
Error  of  Standard  i#        W 

Error  of  Steering-Comp.        2^        W 

Ex.  2, — Comparison  10°  W  andDev.  of 
Standard  20°  E :  Required,  the  Error  of 
the  Compared  Compass. 

Comparison  10°  W 

Dev.  of  Standard  20    E 

Dev.  of  Compared  Comp.  10    E 


Ex.  3. — Comparison  y°.5  E,  and  Error 
of  Standard  i6°.2  E  :  Required,  the  Error 
of  the  Compared  Compass. 

Comparison  7°.5  E 

Error  of  Standard  16  .2  E 

Error  of  Compared  Comp.  23  .7  E 

Ex.  4. — Comparison  i5°,5  E,  and  Error 
of  Standardise  W:  Required,  the  Error 
of  the  Compared  Compass. 

Comparison  I5°-5  W 

Error  of  Standard  9  .o  E 

Error  of  Compared  Comp,    6  .5  W 


B.    COMPASS-COMPAHISONS   ON    SHORE. 

11O.  Comparison  on  Shore:  Detection  of  Local 
Ulagnetism.— It  is  frequently  desirable  to  carry  a  compass  on  shore 
for  [Reciprocal  Observations  with  the  Standard  on  board,  and  for  other 
purposes.  In  such  a  case,  it  will  be  necessary  to  compare  the  com- 
passes, if  any  assurance  of  accuracy  be  desired,  not  only  to  obtain  their 
Constant  Differences  but  to  find  the  Errors  due  to  Local  Magnetism,1  if, 
indeed,  either  exist.  For  this  purpose  we  may  proceed  in  the  following 
manner : 

Rule :  Take  both  compasses  on  shore.  Place  the  Shore-Compass  in 
the  selected  position,  and  the  Standard  Compass  at  a  distance  of  joo  to 
200  feet  from  it.2 

Take  the  Bearing  of  each  from  the  other,  and  note  their  respective 
Eeadiugs.  Again,  place  the  Standard  Compass  about  the  same  dis- 
tance off,  and  about  eight  points  to  one  side  of  the  former  position,  and 
note  the  respective  Eeadiugs.  Finally,  place  the  Standard  about  the 
same  distance  off  and  about  sixteen  points  from  the  first  position,  and 
note  their  Eeadings  as  before. 

Now  compare  these  Bearings,  and  it  will  be  shown — 

First,  if  the  Bearings  of  each  pair  differ  exactly  16  points,  or  180°, 
there  is  neither  Deviation  from  Local  Magnetism  nor  disagreement  of 
Heading  from  any  cause ; 

Secondly,  if  the  difference  between  the  Eeadings  be  more  or  less  than 
1 80°,  but  sensibly  constant,  there  is  probably  no  Deviation  from  Local 
Magnetism,  but  only  a  difference  of  Eeadings  in  the  compasses,  which 
must  be  noted  as  a  correction  to  be  applied  ;  and, 

1  Arising  from  detached  masses  of  iron,  in  sight  or  concealed  from  view,  such  as 
anchors,  guns,  warping-posts,  chains,  etc. ;  or  from  magnetic  iron-ore,  trap  or  basaltic 
rock,  etc. 

3  It  will  be  convenient  to  have  a  tripod  for  the  Standard  Compass,  to  be  moved  about. 
The  other  compass  may  be  set  up  in  its  box  on  a  table,  block,  or  post ;  upon  anything 
of  suitable  height  and  stability. 


12 


90  COMPASS-COMPARISONS. 


Thirdly,  if  the  difference  of  Beadings  is  not  sensibly  the  same,  there 
is  a-  Deviation  from  some  local  cause ;  in  which  case  the  position  selected 
for  the  Compared  Compass  must  be  changed,  in  order  to  iind,  by  an- 
other trial,  if  possible,  a  suitable  position  for  a  station  that  shall  be  free 
from  Local  Error.  Selecting  a  second  position,  make  a  similar  series 
of  observations ;  and  thus  proceed  till  a  satisfactory  position  be  found, 
or  the  particular  locality  abandoned  as  altogether  impracticable. 

Remark. — The  necessity  for  comparing  compasses,  under  the  cir- 
cumstances supposed  in  this  Article,  as  a  preparation  for  the  observa- 
tion of  Serial  Compass-Errors,  should,  we  think,  very  seldom  occur  in 
practice.  As  already  indicated,  it* is  far  better,  in  the  Method  by 
Reciprocal  Bearings,  to  employ  for  the  shore-observations  such  an  an- 
gular instrument  as  a  Theodolite  or  Dumb-Compass,  in  which  any  use 
of  the  Magnetic  Needle  or  Card  is  dispensed  with  ;  not  only  as  being  in- 
trinsically a  more  reliable  instrument,  but  as  relieving  from  all  anxiety 
and  embarrassment  relative  to  the  possible  influence  of  Local  Mag- 
netism. Otherwise,  it  may  often  be  far  better  to  make  use  of  Solar 
Azimuths. 


TABLE  I. 


Compass  Points  and  their  Equivalents  in  Degrees  and  parts  thereof. 

! 

Degree  Equivalents  to  the 

Name  of  Point.           No. 

nearest  — 

No.  i        Name  of  Point. 

Second.     Hundreth.  Tenth. 

o          /         //    ,                     0         j                0 

j 

,  North—  East  or  West.       0 

o      o      o               o.oo           o.o 

0      South—  West  or  East. 

i 

I       24      22                       I.4O                 1.4 

i 

'  i 

2    48    45               2.81            2.8 

i 

I 

4     13       7                4.22  |         4.2 

f  • 

i 

5     37    30               5-63            5-6 

* 

f 

7       i     52               7-03 

7-o 

'*  1 

f 

8    26     15               8.44           8.4 

f 

n_ 

S 

9    50    37  ;             9-84           9-8 

1 

N.  by  E.  or  N.  by  W.         1 

ii    15      o            11.25         11.3 

1 

S.  by  W.  or  S.  by  E. 

i 

12     39    22              12.66          12.7 

| 

14      3    45              14.06          14.1 

JL 

1 

15    28      7              15.47          15-5 

1 

i  i 

16    52    30              16.88          16.9 

•§ 

1 

18     16    52              18.28          18.3 

1 

m: 

i 

19    41     15              19.69          19.7 

1 

1 

21       5     37  ,          21.09          2i.  i 

i 

N.N.E.  or  N.N.  W.               2 

22    30      o             22.50          22.5 

2      S.S.W.  or  S.S.E. 

i 

23     54    22             23.90          23.9 

i 

j. 

4 

25     '8    45             25.31  ;       25.3 

i 

1 

26    43       7             26.72  :       26.7 

1 

i 

28      7    30             28.13          28.1 

1 

1 

29    31     52             29.53          29.5 

i  : 

i 

30    56     15              30.94          30.9 

1      , 

;    » 

32    20    37             32.34         32.3 

i 

N.E.byN.orN.W.byN.     3 

33    45      o             33.75  •       33.8 

3      S.W.byS.orS.E.byS. 

i 

35      9    22             35.16  •       35.2 
36    33    45             36-56          36-6 

**] 

i 

37    5*      7             37-97         38-° 

i 

i 

39    22    50             39.38         39.4 

i 

t 

40    46    52             40.78  ;       40.8 

*  i 

;  I 

42     ii     15             42.19         42.2 

f 

1 

43    35     37             43-59  j       43-6 

i 

i  N.E.  or  N.  W.                         4 

45      o      o             45-°o          45-° 

4      S.W.  or  S.E. 

:          i 

46      24      22                  46.40   ;          46.4 

i  ' 

47    48    45             47-8i  \       47-8 

1 

49     13       7             49.22          49.2 

1 

1 

50    37    30             50.63 

50.6 

i 

f 

52     i    52  i         52-03 

52-0 

i 

f 

53    26    i5             53-44 

53-4 

f 

1 

54    5°    37  ;           54.84 

54-8 

I 

N.E.byE.orN.W.byW.!    5 

56    15      o             56.25 
57    39    22             57-66 

56-3 

57-7 

5 

i 

S.W.byW  orS.E.byE. 

!  i 

59      3    45  !           59-06 

59-i 

i 

f 

60    28      7             60.47 

60.5 

f 

\                                   '  * 

61     52    30             61.88 

61.9 

f 

63     16    52             63.28 

63-3 

1 

i  1 

'      •  1 

64   41    15  :         64.69 
66      5    37  1          66.09 

64.7 
66.1 

1 

E.N.E.  or  W.N.W.              6 

67    30      o  i           67.50 

67.5 

6 

W.S.W.  or  E.S.E. 

i 

68    54    22  j           68.90          68.9 

i 

i 

70    1  8    45  !           70.31  |       70.3 

i 

1 

71     43       7  i           71.72 

7i-7 

1 

i 

73      7    3°  i           73-13 

73-i 

1 

i 

74    31     52  !           74-53 

74-5 

t 

f 

75    56     15  '           75-94 

75-9 

1 

1 

77    20    37  ,           77.34 

77-3 

1 

E.  by  N.  or  W.  by  N.         7 

78    45      o  |           78.75 

78.8 

7 

W.  by  S.  or  E.  by  S. 

i 

80      9    22  i           80.  1  6 

80.2 

i 

81     33    45  !           81.56 

81.6 

i 

f 

82    58      7  i           82.97 

83.0 

± 

84      22      30   !               84.38 

84-4 

| 

1 

85   46   52  !        85.78 

85.8 

| 

1 

87    II    15  1         87.19 

87.2 

f 

i 

88    35    37  i           88.59 

88.6 

1 

East  or  West.                      8 

90      o      o              90.00 

90.0 

8 

West  or  East. 

TABLE  II. 


TABLE  III. 


TABLE  IV. 


3 


Mean  Solar  into  Sidereal  Time. 

Sidereal  into  Mean  Solar  Time. 

Solar         .HH 
Hours.       Add' 

Solar 
Miu. 

Add. 

Solar 
Sec. 

Add. 

Sid. 
Hours. 

Subtract, 

Sid. 

Miu. 

Subtract. 

Sid.    * 
Sec.    J 

iubtract. 

m      s 

s 

s 

m       s 

s 

1 

s 

1         o    9.86 

1 

0.16 

1 

o.oo 

1 

o    9.83 

1 

0.16 

1   |      o.oo 

2        o  19.71 

2          0.33 

2          o.oo 

2 

o  19.66 

2 

°-33 

2  1      o.oo 

3        o  29.57 

3          0.49 

3          o.oi 

3 

o  29.49 

3 

0.49 

3  i      o.oi 

4        o  39.43 

4          0.66 

4          o.oi 

4 

o  39.32 

4 

0.66 

4         o.oi 

5        o  49.28 

5          0.82 

5          o.oi 

5 

o  49.15 

5 

0.82 

5 

O.OI 

6        o  59.14 

6          0.98 

6              O.O2 

6 

o  58.98 

6 

0.98 

6 

0.02 

7             9.00 

7          1-15 

7    !          0.02 

7 

8.8  1 

7 

1-15 

7  I      0.02 

8            18.85 

8          1.31 

8              O.O2 

8 

18.64 

8 

P-3' 

8         0.02 

9           28.71 

9          1.48 

9              0.02 

9 

28.47 

9 

1.47 

9    !         0.02 

!    10           38-56 

1O          1.64 

1O          0.03 

10 

38-30 

10 

1.64 

1O         0.03 

11           4842 

11          i-Si 

1  1        0.03 

11 

48.13 

11 

i.  80 

11       0.03 

12           58.28 

12        •  1.97 

12          0.03 

12 

57-95 

12 

1.97 

12  <      0.03 

13        2    8.13 

13          2.13 

13          0.04 

13 

2     7.78 

13 

2.13 

13  '•      0.04 

14        2  17.99 

14          2.30 

14          0.04 

14 

2    I7.6l 

14 

2.29 

14         0.04 

15        2  27.85 

15          2.46 

15          0.04 

15 

2    27.44 

15 

2.46 

15         0.04 

16        2  37.70 

16    :         2.63 

16          0.04 

16 

2   37-27 

16 

2.62 

16  i      0.04 

17            2   47.56 

17  !       2.79 

17          0.05 

17 

2   47.10 

17 

2.78 

17  :      0.05 

18        2  57.42 

18   !       2.96 

18          0.05 

18 

2    56.93 

18 

2-95 

18         0.05 

19        3    7-27 

19  !       3-12 

19          0.05 

19 

3    6.76 

19 

3-11 

19        0.05 

20        3  17-13 

20          3-29 

20          0.05 

20 

3  16.59 

20 

3.28 

2O        0.05 

21        3  26.99 

21          345 

21          0.06 

21 

3  26.42 

21 

3-44 

21         0.06 

22  i     3  36.84 

22  :       3-61 

22 

0.06 

22 

3  36-25 

22 

3-60 

22         0.06 

23        3  46-70 

23 

3-78 

23 

0.06 

23 

3  46.08 

23 

3-77 

23         0.06 

24        3  56.56 

24 

3-94 

24 

0.07 

24 

3  55-91 

34 

3-93 

24         0.07 

25 

4.11 

25 

0.07 

25 

4.10 

25 

0.07 

26 

4-27 

26 

0.07 

26 

4.26 

26 

0.07 

27 

4-44 

27 

0.07 

27 

4.42 

27 

O.O7 

28 

4.60 

28 

0.08 

28 

4-59 

28 

0.08 

29 

4-76 

29 

0.08 

29 

4-75 

29 

0.08 

3O 

4-93 

30 

0.08 

30 

4.91 

30 

0.08 

31 

5-09 

31 

0.08 

31 

5.08 

31 

0.08 

32 

5-26 

32 

0.09 

32 

5-24 

32 

O.O9 

33 

5.42 

33 

0.09 

33 

5-41 

33 

O.O9 

34 

5-59 

34 

0.09 

34 

5-57 

34 

0.09 

35 

5-75 

35 

0.10 

35 

5-73 

35 

O.IO 

36 

5-91 

36 

0.10 

36 

5-9° 

36 

O.IO 

37 

6.08 

37 

O.IO 

37 

6.06 

37 

O.IO 

38 

6.24 

38 

O.I  I 

38 

6.23 

38 

O.II 

39 

6.40 

39 

O.I  I 

39 

6-39 

39 

O.II 

40 

6.57 

40 

O.II 

40 

6-55 

40 

O.II 

41 

6.74 

41 

O.I  I 

41 

6.72 

41 

O.II 

42 

6.90 

42 

0.12 

42 

6.88 

42 

0.12 

43 

7.06 

43 

0.12 

43 

7.04 

43 

O.I2 

44 

7-23 

44 

0.12 

44 

7.21 

44 

0.12 

45 

7-39 

45 

0.12 

45 

7-37 

45 

0.12 

46 

7.56 

46 

0.13 

46 

7-54 

46 

O.I3 

47 

7-72 

47 

0.13 

47 

7.70 

47 

0.13 

48 

7.89 

48 

0.13 

48 

7.86 

48 

O.I3 

49 

8.05 

49 

0.14 

49 

8.03 

49 

0.14 

50 

8.21 

50 

0.14 

5O 

8.19 

50 

0.14 

51 

8.38 

51 

O.I4 

51 

8.36 

51 

0.14 

52 

8-54 

52 

0.14 

52 

8.52 

52 

O.I4 

53 

8.71 

53 

0.15 

53 

8.68 

53 

0.15 

54 

8.87 

54 

0.15 

54 

8.85 

54 

O.I5 

55 

9.04 

55 

0.15 

55 

9.01 

55 

0.15 

56 

9.20 

56 

0.15 

56 

9.17 

56 

0.15 

57 

9-36 

57 

0.16 

57 

9-34 

57 

0.16 

58 

9-53 

58 

0.16 

58 

9-5° 

58 

0.16 

59 

9.69 

59 

0.16 

59 

9.67 

59 

0.16 

60 

9.86 

60 

0.16 

60 

9-83 

60 

0.16 

TABLE  V. 


Length  of  a  Degree  in  Latitude  or  Longitude. 

Lat. 

Deg.  of  Long. 

Deg.  of  Lat. 

Lat. 

Deg.  of  Long. 

Deg.  of  Lat. 

0 

Stat.  Miles. 

Naut.  Miles. 

Stat.  Miles. 

Naut  Miles. 

o 

Stat.  Miles. 

Naut.  Miles. 

Stat.  Miles 

Naut.  Miles. 

0 

69.160 

60.000 

68.698 

59.600 

45 

48.986 

42.498 

69.044 

59.899 

1 
2 

.150 

.119 

59-991 
•964 

.698               .600 
.699     i          .601 

46 
47 

.126 

47.251 

4L752 
40.993 

.056 
.068 

.910 
.920 

3 

.066 

.919 

.700     '          .602 

48 

46.362 

.222                  .080 

-931 

4 

68.992 

.855 

.702     '          .603 

49 

45-459 

39-439 

.092 

.941 

5 
6 

68.898 
.783 

59-773 
.673 

68.704 
.706 

59.605 
.607 

50 
51 

44.542 
43.611 

38-643 
37.835 

69.104 
.116 

59-951 
.962 

7 

•647 

•555 

•709 

.609 

52 

42.667 

.016 

.128 

8 

.491 

.419 

.712 

.612 

53 

41.710 

36.186 

.140 

.982 

9 

•3H 

.265 

.715 

.615 

54 

40.740 

35-344 

•I5I 

.992 

10 
11 

68.116 
67.898 

59-093 
58.904 

68.719 
-723 

59.618 
.621 

55 
56 

39.758 
38.763 

34-491 
33.628 

69.162 

60.002 

.012 

12 

•659 

.697 

.728 

.625 

57 

37-756 

32-755 

.184 

.022 

13 

.400 

•472 

•733 

.629 

58 

36.737 

31.872 

•195 

.032 

14 

.120 

.229 

•738 

-634 

59 

35-707 

30.979 

.206 

.041 

15 

66.820 

57.968 

.  68.744 

59.639 

60 

34.666 

30.076 

69.217 

60.050. 

16 

•499 

.690 

-75° 

.645 

61 

33-615 

29.164 

.228 

.059 

17 
18 

158 
65-797 

aj 

•764 

.651 
.657 

62 
63 

32-553 
31.481 

28.242 
27.311 

.238 
.248 

•077 

19 

.416 

56.751 

.771 

.663 

64 

30.399 

26.372 

.258 

.086 

20 
21 

65.015 
64.594 

56.404 
.039 

6S:W7 

59.669 
.676 

65 
66 

29.308 
28.208 

25-425 
24.471 

69.268 
•277 

60.094 
.102 

22 

•154 

55.657 

•795 

•683 

67 

27.100 

23-509 

.286 

.110 

23 

63-695 

.258 

.804 

.69I 

68 

25.983 

22.540 

•294 

.117 

24 

.216 

54.843 

.813 

.699 

69 

24-857 

21.564 

.302 

.124 

25 

62.718 

54-4" 

68.822 

59.707 

70 

23-723 

20.582 

69.310 

60.131 

26 
27 

.201 
61.665 

53.962 
•497 

.831 
.840 

•715 
.723 

71 

72 

22.582 
21-435 

19-593 
18.598 

.318 
.326 

•137 
•143 

28 

.110 

.016 

•850 

•731 

73 

20.282 

17-597 

•333 

.149 

29 

60.536 

52.518 

.860 

.740 

74 

19.122 

16.590 

•339 

•155 

30 

59-944 

52.005. 

68.870 

59-749 

75 

17.956 

I5-578 

69.345 

60.161 

31 

32 
33 

58706 
.060 

51.476 
50.931 
•370 

.881 
.892 
•903 

.758 
.767 
•776 

76 
77 

78 

15-607 
14.425 

14.561 
13-539 
12-513 

•351 

•357 

r-  -362 

.166 
.171 
.175 

34 

57.396 

49-794 

.914 

.786 

79 

13-238 

11.484 

.367 

.179 

35 
36 

56.715 
.016 

49-203 
48.597 

68.925 
•936 

59-796 
.806 

80 
81 

12.047 

10.452 
9.417 

69-371 

-375 

60.183 
.186 

37 

55.30° 

47-976 

•947 

.816 

82 

9.656 

8-379 

-378 

.189 

38 

54.568 

•340 

•959 

.826 

83 

8.456 

7.338 

.381 

.192 

39 

46.690 

.971 

.836 

84 

7-253 

6.294 

.384 

.194 

40 
41 
42 

53.053 
52.271 

5M73 

46.026 
45-348 
44-656 

68.983 

*  -995 
69.007 

59.846 
!866 

85 

86 

87 

6.048 
4.841 
3-632 

5.248 
4.200 
3.151 

69.387 
.389 
•390 

60.196 
.198 
.199 

43 

50.659 

43-95° 

.019 

.877 

88 

2.422 

2.IOI 

-391 

.200 

44 

49.830 

.231 

.031 

.888 

89 

1.  211 

1.050 

•392 

.201 

TABLE  VI. 


5 


Logarithms  of  Numbers  and  Small  Arcs. 

N. 

Log. 

N.           Log. 

N.           Log. 

N. 

Log. 

N.            Log. 

0 

2O             3OI° 

40 

6021 

60 

7782 

80 

9031 

1              oooo 

1             3222 

1 

6128 

1 

7853 

t 

9085 

2  \            3010 

2             3424 

2 

6232 

2 

7924 

2 

9138 

3 

4771 

3  |          3617 

3 

6335 

3 

7993 

3 

9191 

4 

6021 

4  L         3802 

4 

6435 

4 

8062 

4 

9243 

5 

6 

6990 

7782 

5  j          3979 
6  |          415° 

5 
6 

6532 
6628 

5 

6 

8129 
8i95 

5 
6 

9294 
9345 

7 

8451 

7  |          43H 

7 

6721 

7 

8261 

7 

9395 

8 

9031 

8             4472 

6812 

8 

8325 

8 

9445 

9 

9542 

9             4624 

9 

6902 

9 

8388 

9 

9494 

10 

0000 

30             477i 

50 

6990 

7O 

8451 

90 

9542 

1 

2 

3 

0414 
0792 

"39 

1 

1   1          49H 
2  !          5052 
»  !          5185 

1 
2 
3 

7076 
7160 
7243 

1 

2 

3 

8513 
8573 
8633 

1 
2 
3 

9590 
9638 

9685 

4 

1461 

4  |          5315 

4 

7324 

4 

8692 

4 

973  1 

5 
6 

1761 

2041 

5             5441 
«            5563 

5 
6 

7404 
7482 

5 
6 

8751 
8808 

5 
6 

9777 
9823 

7 

* 
9 

2304 

2553 
2788 

7             5682 
8             5798 
9             5911 

7 

§ 
9 

7559 
7634 
7709 

7 
8 
9 

8865 
8921 
8976 

7 

8 
9 

9868 
9912 
9956 

20 

3010 

4O            6021 

6O 

7782 

80 

9031 

too 

oooo 

N. 

Log. 

N.           Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 

Small  Arcs.                                                            Small  Arcs. 

s      h   m     s                                                              s       h    TO     s 

II             Q           1             II                                                                                                                                          II             Q            1            II 

0  =  000                                                           50  =  o    o    50 

10  =  00      10                                                                                      60=00 

2O  •=  O      O      2O                                                                                      7O  =:  O              IO 

30  .  =  o    o    30                                                          So  =  o         20 

40  =  o    o    40                                                          90  =  o         30 

50  =  o    o    50                                                         loo  =  0         40 

TABLE  VI 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

O 

1 

2 

3 

4 

5 

6 

7 

8 

9 

100 

oooo 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035    0039 

1 
2 
3 
4 
5 
6 
7 
8 
9 

0043 
0086 
0128 
0170 

O2I2 
0253 
0294 
0334 
0374 

0048 
0090 
OI33 

oi75 
0216 

0257 
0298 

0338 
0378 

0052 
0095 
0137 
0179 

0220 
026l 
0302 
0342 
0382 

0056 
0099 
0141 
0183 
0224 
0265 
0306 
0346 
0386 

0060 
0103 
0145 
0187 
0228 
0269 
0310 

0350 
0390 

0065 
OIO7 
0149 
0191 

0233 
0274 

03H 
0354 
0394 

0069 

OIII 

0154 

oi95 

0237 
0278 
0318 
0358 
0398 

0073 
0116 
0158 
0199 
0241 
0282 
0322 
0362 
0402 

0077 

OI2O 
Ol62 
O2O4 
0245 
0286 
0326 
0366 
0406 

0082 
0124 
0166 
0208 
0249 
0290 

0330 
0370 
0410 

0449 

110 

0414 

0418 

0422 

0426 

0430 

0434 

0438 

0441 

0445 

1 
2 
3 
4 
5 
6 
7 
8 
9 

0453 
0492 

Q531 
0569 
0607 
0645 

0682 
0719 
0755 

0457 
0496 

°535 

0573 
0611 
0648 
0686 
0723 
0759 

0461 
0500 
0538 

0577 
0615 
0652 
0689 
0726 
0763 

0465 
0504 
0542 
0580 
0618 
0656 
0693 
°730 
0766 

0469 
0508 
0546 
0584 
0622 
0660 
0697 

0734 
0770 

0473 
0512 

°55° 
0588 
0626 
0663 
0700 

0737 
0774 

0477 
05*5 
0554 
0592 
0630 
0667 
0704 
0741 
0777 

0481 
0519 
0558 
0596 
0633 
0671 
0708 

0745 
0781 

0484 
0523 
0561 

0599 
0637 
0674 
O7II 
0748 
0785 

0488 
0527 
0565 
0603 
0641 
0678 

0715 

0752 
0788 

I  120 

0792 

0795 

0799 

0803 

0806 

0810 

0813 

0817 

0821 

0824 

1 
2 
3 
4 
5 
6 
7 
8 
9 

0828 
0864 
0899 

0934 
0969 
1004 
1038 
1072 
1106 

0831 
0867 
0903 
0938 

0973 
1007 

1041 

'075 
1109 

0835 
0871 
0906 
0941 
0976 
IOII 

1045 
1079 
III3 

0839 
0874 
0910 

0945 
0980 
1014 
1048 
1082 
1116 

0842 
0878 
0913 
0948 
0983 
1017 
1052 
1086 
1119 

0846 
0881 
0917 
0952 
0986 

IO2I 

1055 
1089 

"23 

0849 
0885 
0920 

0955 
0990 
1024 
1059 
1092 
1126 

0853 
0888 
0924 

0959 
0993 
1028 

1062 
1096 
1129 

0856 
0892 
0927 
0962 
0997 
1031 
1065 
1099 

"33 

0860 
0896 
0931 
0966 

1000 

I035 
1069 
1103 
1136 

130 

"39 

"43 

1146 

"49 

"53 

1156 

"59 

1163 

1166    1169 

1 
2 
3 
4 
5 
6 
7 
8 
9 

"73 
1206 
1239 
1271 
1303 
1335 
1367 
1399 
143° 

1176 
1209 
1242 
1274 
I3°7 
1339 
1370 
1402 
H33 

"79 

1212 

1245 

1278 
I3IO 
1342 
1374 
1405 
H36 

1183 
1216 
1248 
1281 
1313 
1345 

1377 
1408 
1440 

1186 
1219 
1252 
1284 
1316 
1348 
1380 
1411 
1443 

1189 

1222 

1255 
1287 

I3J9 
I35i 

1383 
1415 
1446 

"93 
1225 
1258 
1290 
1323 
1355 
1386 
1418 
1449 

1196 
1229 
1261 
1294 
1126 

1358 

1389 
1421 

H52 

1199      I2O2 

1232   1235 
1265   1268 
1297    1300 
1329   1332 
1361    1364 
1392    1396 
1424    1427 
1455   1458 

140 

1461 

1464 

1467 

1471 

H74 

H77 

1480 

1483 

1486 

1489 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1492 
1523 
1553 

1584 
1614 
1644 
1673 
1703 
1732 

1495 
1526 

1556 

1587 
1617 
1647 
1676 
1706 
1735 

I498 
1529 
1559 
1590 
l62O 
1649 
1679 
1708 
1738 

1501 
1532 
1562 

1593 
1623 
1652 
1682 
1711 
1741 

i5°4 
1535 
1565 
'596 
1626 
1655 
1685 
1714 
1744 

1508 

1538 
i569 

1599 
1629 
1658 
1688 

1717 
1746 

15" 
1541 
1572 
1602 
1632 
1661 
1691 
1720 
1749 

I5H 

1544 
1575 
1605 

1635 
1664 

1694 
1723 
1752 

1517 

1547 
1578 
1608 
1638 
1667 
1697 

1726 

1755 

1520 
1550 
1581 
1611 
1641 
1670 
1700 
1729 
1758 

150 

1761 

1764 

1767 

1770 

1772 

1775 

1778 

1781 

1784 

1787 

s   h  TO  s                           s   h  m   s 

II     Q    1     II                                               II     0     t     II 

100  =  o  i  40                       1000  =  o  16  40 
no  =  o  i  50                       iioo  —  o  18  20 

J2O  =  O2    O                                12OO  —  O   2O    O 
130  =  0   2   10                                1300  =  0   21   40 
J4O  =  O   2   2O                                I4OO  —  O   23   2O 
150  =  0   2   30                                1500  =  0   25    0 

TABLE  VI. 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

0 

1 

% 

3 

4 

5 

6   |   » 

8 

9 

150 

1761 

1764 

I767 

1770 

1772 

1775 

1778    1    I78l       1784 

1787 

1 

9 
3 
4 
£ 
6 

!   * 
§ 
9 

1790 

1818 
1847 
1875 
1903 
193  i 

1959 
1987 
2014 

1793 
1821 
1850 
1878 
1906 
1934 
1962 
I989 
2OI7 

1796 
1824 
1853 

1881 
1909 
1937 

i965 
1992 
2019 

1798 
1827 
1855 
1884 
1912 
1940 
1967 
1995 

2O22 

1801 
1830 
1858 
1886 

1915 
1942 

1970 
1998 
2025 

1804 

1833 
1861 

1889 
1917 
!945 
1973 

2OOO 
2028 

1807    1810 
1836    1838 
1864  j  1867 
1892  |  1895 
1920  !  1923 
1948  i  1951 
1976    1978 
2003  :  2006 
2030    2033 

1813 

1841 
1870 
1898 
1926 
1953 
1981 
2009 
2036 

1816 
1844 
1872 
1901 
1928 
1956 
1984 

201  1 
2038 

i  160 

2041    2044 

2047 

2049 

2052 

2055 

2057 

2060 

2063 

2066 

2092 
2119 
2146 
2172 

2198 
2225 
225I 
227O 
2302 

1 

9 
3 
4 
5 
6 
7 
8 
9 

2068 
2095 

2122 
2148 
2J75 
2201 
2227 

2253 
2279 

2305 

2071 
2098   | 
2125 
2I5I 
2177 
22O4 
2230 
2256 
228l 

2074 

2101 
2I27 

2154 
2180 
2206 
2232 
2258 
2284 

2076 
2103 
2130 
2156 
2l83 
2209 
2235 
226l 
2287 

2079 
2106 
2i33 

2159 
2185 

2212 
2238 
2263 
2289 

2O82 
2109 
2135 
2l62 
2188 
2214 
2240 
2266 
2292 

2084 

2III 
2I38 
2164 
2191 
2217 
2243 
2269 
2294 

2087 
2II4 
2I4O 
2167 
2193 
2219 
2245 
2271 
2297 

2090 
2117 
2143 
2170 
2196 

2222 
2248 

2274 
2299 

170 

2307 

2310 

2312 

2315 

23!7 

2320 

2322 

2325 

2327   ! 

1 
2 
3 
4 
5 
6 
7 
8 
9 

2330 

2355 
2380 

2405 
2430 
2455 
2480 
2504 
2529 

2333 
235* 
2383 
2408 

2433 
2458 

2482 
2507 
2531 

2335 
2360 

2385 
2410 

2435 
,  2460 

2485 
2509 

2533 

2338 
2363 
2388 

2413 
2438 
2463 

2487 
25I2 

2536 

2340 

2365 
2390 

2415 

2440 
2465 
2490 
25H 
2538 

2343 
2368 

2393 
2418 

2443 
2467 

2492 
2516 
2541 

2345 
2370 
2396 
2420 

2445 
2470 

2494 
2519 

2543 

2348 
2373 
2398 

2423 
2448 
2472 
2497 
2521 
2545 

2350 
2375 
2401 

2425 
2450 
2475 
2499 
2524 
2548 

2353   , 
2378   | 
2403   i 

2428 

2453  ! 

2477  ; 
2502 
2526  j 

2550  ! 

180 

2553 

2555 

2558 

2560 

2562 

2565 

2567 

2570 

2572 

2574  | 

1 
2 
3 

4 

5 
6 
7 

8 
9 

2577 
2601 
2625 
2648 
2672 
2695 
2718 
2742 
2765 

2579 
2603 
2627 
2651 
2674 
2697 
2721 

2744 
2767 

2582 
2605 
2629 

2653 
2676 
2700 
2723 
2746 
2769 

2584 
2608 
2632 

2655 
2679 
2702 
2725 
2749 
2772 

2586 
26lO 
2634 
2658 
2681 
2704 
2728 

2751 

2774 

2589 
2613 
2636 
2660 
2683 
2707 
2730 

2753 
2776 

2591 
2615 
2639 
2662 
2686 
2709 
2732 

2755 
2778 

2594 
2617 
2641 
2665 
2688 
2711 

2735 
2758 
278! 

2596 
2620 
2643 
2667 
2690 
27H 

2737 
2760 

2783 

2598  i 
2622 
2646  i 
2669  : 
2693 
2716 

2739  1 
2762  ; 
2785  j 

19O 

2788 

2790 

2792 

2794 

2797 

2799 

2801 

2804 

2806 

2808  ! 

1 

2 
3 
4 

5 
6 
7 

8 
9 

28lO 

2833 
2856 

2878 
29OO 
2923 

2945 
2967 
2989 

2813 

2835 

2858 

2880 
2903 
2925 

2947 
2969 
2991 

2815 
2838 
2860 
2882 
2905 
2927 
2949 
2971 
2993 

2817 
2840 
2862 
2885 
2907 
2929 
2951 
2973 
2995 

28l9 

2842 
2865 
2887 
2909 
2931 

2953 
2975 
2997 

2822 
2844 
2867 
2889 
2911 
2934 
2956 
2978 
2999 

2824 
2847 
2869 

2891 
2914 
2936 

2958 
2980 
3OO2 

2826 
2849 
2871 

2894 
2916 
2938 
2960 
2982 
3004 

2828 
2851 
2874 
2896 
2918 
2940 
2962 
2984 
3006 

2831 

2853  ! 
2876 

2898 
2920 
2942 
2964 
2986 
3008 

2OO 

3OIO 

3012 

3°I5 

3017 

30I9 

3021 

3023 

3°25 

3028 

3030 

s   h  m   s                           s   h  m   s 

II      O    1     II                                                   II     V      1      11 

150  =  0   2   30                                1500  =  0   25    0 
l6o  =:  O   2   40                                  I6OO  =  O   26   40 
1  70  =  O   2   50                                  1  7OO  =  O   28   2O 

180  =  03   o                      1800  =  o  30   o 
190  =  0  3  10                      1900  =  0  31  40 
200  =  o  3  20                      2000  =  o  33  20 

8 


TABLE  VI. 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

O 

1 
3012 

2     3 

4 

5 

6 

7 

8 

«* 

\  200 

"IT 

2 
3 
4 

5 
6 
7 

8 
9 

3010 

3015 

3oi7 

3019 

3021 

3023 

3025 

3028 

3030  i 

3032  i  3034 
3°54    3°56 
3°75  ;  3°77 
3096    3098 
3118    3120 
3139  •  3Hi 
3160  i  3162 

3181    3183 
3201    3204 

3036 
3058 
3079 
3101 
3122 
3143 
3164 
3185 

3206 

3038 
3060 
3081 

3I03 
3124 
3H5 
3166 

3187 
3208 

3°4i 
3062 
3084 

3^5 
3126 

3H7 
3168 

3189 
3210 

3043 
3064 
3086 

3*07 
3128 

3149 
3170 

3*9i 
3212 

$& 

3088 
3109 
3!3° 
3151 
3172 
3193 
3214 

3°47 
3069 
3090 
3111 
3132 
3153 
3'74 
3'95 
3216 

3°49 
3071 
3092 

3"3 
3U4 
3156 
3176 

3197 
3218 

3051  ; 
3°73 
3°94 

3"5 
3*37 

3158  ! 

3179 
3199 
3220  ' 

21O 

3222 

3224 

3226 

3228 

323° 

3233 

3235 

3237 

3239 

3241  | 

1 
2 
3 
4 
5 
6 
7 
8 
9 

3243 
3263 
3284 

3304 
3324 
3345 
3365 
3385 
3404 

3245 
3265 
3286 

3306 
3326 
3347' 
3367 
3387 
3406 

3426 

3247 
3267 
3288 

3308 
3328 

3349 
3369 
3389 
3408 

3249 
3270 
3290 

33io 
3330 
3351 
337i 
3391 
34io 

3251 
3272 
3292 
3312 
3332 
3353 
3373 
3393 
3412 

3253 
3274 
3294 
33H 
3334 
3355 
3375 
3395 
34H 

3255 
3276 
3296 
33i6 
3336 
3357 
3377 
3397 
34i6 

3257 
3278 
3298 

33i8 
3339 
3359 
3379 
339| 
34i8 

3259 
3280 
3300 
3320 
334i 
336i 

338i 
3400 
3420 

3261 
3282  | 
3302  i 
3322 
3343 
3363 

3383 
3402 
3422 

22O 

3424 

3428 

343° 

3432 

3434 

3436 

3438 

3440 

3442 

1 
2 
3 
4 
5 
6 
7 
8 
9 

3444 
3464 
3483 
35°2 
3522 
354i 

3560 
3579 
3598 

3446 
3465 
3485 
3504 
3524 
3543 
3562 
358i 
3600 

3448 
3467 
3487 
35o6 
3526 
3545 
3564 

3602 

3450 
3469 
3489 
3508 
3528 
3547 
3566 

3585 
3604 

3452 
347i 
349i 
35io 
353° 
3549 
3568 

3I8Z 
3606 

3454 
3473 
3493 
3512 
3531 
3551 
3570 

£ 

3456 
3475 
3495 
35H 
3533 
3553 
3572 
359i 
3610 

3458 
3477 
349*7 
35i6 
3535 
3555 
3574 
3593 
3612 

3460 
3479 
3499 
35i8 
3537 
3556 
3576 
3595 
3614 

3462 
348i 
3501 
3520 
3539 
3558 

3577 
3596 
3615 

230 

3617 

3619 

3621 

3623 

3625 

3627 

3629 

3630 

3632 

3634 

1 
2 
3 
4 
5 
6 
7 
8 
9 

3636 
3655 
3674 
3692 
37" 
3729 

3747 
3766 
3784 

3638 
3657 
3675 
3694 
3713 
3731 

3749 
3768 

3786 

3640 
3659 
3677 
3696 
37H 
3733 
375i 

3788 

3642 
3660 
3679 
3698 
3716 
3735 
3753 

3789 

3644 
3662 
3681 

37oo 
37i8 
3736 
3755 
3773 
379i 

3646 
3664 
3683 
3701 
3720 
3738 

3757 
3775 
3793 

3647 
3666 

3685 

3703 
3722 
3740 
3758 
3777 
3795 

3649 
3668 

3687 
3705 
3724 
3742 
3760 
3779 
3797 

3651 

$8 

3707 
3725 
3744 
3762 
378o 
3798 

3653 
3672 
3690 

3709 
3727 
3746 

3764 
3782 
3800 

3818 

240 

3802 

3804 

3806 

3808 

3809 

3811 

3813 

3815 

38i7 

1 
2 
3 
4 
5 
6 
7 
8 
9 

3820 
3838 
3856 

3|74 
3892 
3909 
3927 
3945 
3962 

3822 
3840 
3858 
3876 
3893 
39" 

3929 
3946 
3964 

3824 
3842 
3860 

3877 
3895 
3913 

393° 
3948 
3965 

3826 
3844 
3861 

3879 
3897 
3915 
3932 
3950 
3967 

3827 
3f45 
3863 
3881 

3*99 
3916 

3934 
3952 
3969 

3829 
3847 
3865 

3883 
3901 
39i8 

3936 
3953 
3971 

3831 
3849 
3867 

3885 
3902 
3920 
3938 
3955 
3972 

3833 
3851 
3869 

3886 

3904 
3922 

3939 
3957 
3974 

3f35 
3852 
3870 

3888 
3906 
3923 
3941 
3959 
3976 

.3*36 
3854 
3872 
3890 
3908 
3925 

3943 
3960 

3978 

250 

3979 

398i 

3983 

3985 

3986 

3988  |  3990  |  3992 

3993 

3995 

s   h  m   s                           s   h  m   s 

1.      Q    1     II                                                   II      0     /     // 

200  =  0   3   20                                2000  —  0   33   2O 

210  =  o  3  30                      2100  =  o  35   o 

22O  =  O   3   40                                22OO  =  O   36  40 
230  =:  O   3   5O                                  23OO  =  O   38   2O 

240  =  04   o                       2400  =  o  40   o 
250  =  o  4  10                       2500  =  0  41  40 

TABLE  VI. 


j 

O 

Logarithm 

us  of  Numbers  and  Small  Ares. 

1  No. 

1 

2 

3 

4 

5 

6789 

250 

3979    398i 

3983 

3985   3.986 

3988 

3990 

3992 

3993  i   3995 

1 

2 
3 
4 
5 
6 
7 

9 

3997 
4014 

403  * 
4048 
4065 
4082 
4099 
4116 
4133 

3998 
4016 

4033 
4050 
4067 
4084 
4101 
4118 

4000 
4017 
4035 
4052 
4069 
4086 

4103 
4120 
4136 

4002 
4019 
4036 

4053 
4071 
4087 
4104 
4121 
4138 

4004 
4021 

4038 

4055 
4072 
4089 
4106 
4123 
4140 

4005 
4023 
4040 

4057 
4074 
4091 
4108 
4125 
4141 

4007 
4024 
4041 

4059 
4076 

4093 
4109 
4126 
4H3 

4009 
4026 
4043 
4060 
4077 
4094 
4111 
4128 
4H5 

4011 
4028 
4045 
4062 
4079 
4096 

4"3 
413° 
4146 

4012 
4029 
4047 
4064 
4081 
4098 
4115 

4131 
4148 

260 

4150 

4151 

4153    4155    4156 

4158 

4160 

4161    4163 

4165 

1 

3 
4 

5 
6 
7 

8 
9 

4166 

4183 
4200 

4216 

4232 
4249 

4265 
4281 
4298 

4168 

4185 
4201 

4218 

4234 
4250 

4267 
4283 
4299 

4170 
4186 
4203 
4219 
4236 
4252 
4268 
4285 
43°  l 

4171 

4188 
4205 
4221 
4237 
4254 
4270 
4286 
4302 

4173 
4190 

4206 

4223 
4239 
4255 
4272 
4288 
4304 

4175 
4191 
4208 
4224 
4241 
4257 

4273 
4289 
4306 

4176 

4193 
4209 

4226 
4242 
4259 

4275 
4291 

43°7 

4178 

4195 
4211 

4228 

4244 
4260 

4276 
4293 
43°9 

4180 
4196 
4213 
4229 
4246 
4262 
4278 
4294 

4i8r 
4198 
4214 

4231 
4247 
4263 

4280 
4296 
4312 

270 

43Z4 

43i5 

4317 

4318 

4320 

4322 

4323 

4325 

4326 

4328 

1 
2 
3 
4 
5 
6 
7 

9 

433° 
4346 
4362 

4378 

4393 
4409 

4425 
4440 

4456 

433  i 
4347 
4363 
4379 
4395 
44" 
4426 
4442 
4458 

4333 
4349 
4365 
438i 
4396 
4412 

4428 
4444 
4459 

4335 

4366 
4382 
4398 
4414 

443° 
4445 
4461 

4336 
4352 
4368 

4384 
4400 

4415 

443  * 
4447 
4462 

4338 
4354 
4370 

4385 
4401 

4417 

4433 
4448 
4464 

4339 

4355 
437i 
4387 

4403 
4419 

4434 
445° 
4465 

4341 
4357 
4373 

4389 
4404 
4420 

4436 
445i 
4467 

4342 
4358 
4374 
4390 
4406 
4422 

4437 
4453 
4468 

4344 
4360 

4376 

4392 
4408 
4423 
4439 
4454 
4470 

280 

4472 

4473 

4475 

4476 

4478 

4479 

4481 

4482 

4484 

4486 

1 
2 
3 
4 
5 
6 
7 

9 

4487 
4502 
4518 

4533 

4564 
4579 
4594 
4609 

4489 
4504 
4519 
4535 
4550 
4565 
4580 

4595 
4610 

4490 
4506 
4521 
4536 
4552 
4567 
4582 
4597 
4612 

4492 

45°7 
4522 

4538 
4553 
4568 

4583 
4598 
4613 

4493 
45°9 
4524 
4539 
4555 
4570 

4600 
4615 

4495 

4526 

4541 
4556 
457i 
4586 
4601 
4616 

4496 
45  12 
4527 
4542 
4558 
4573 
4588 
4603 
4618 

4498 
4513 
4529 
4544 
4559 
4574 

4589 
4604 
4619 

4499 
4515 
4530 
4545 

4576 

459i 
4606 
4621 

45°  i 

4532 

4547 
4562 
4577 

4592 
4607 
4622 

290 

4624 

4625 

4627 

4628 

4630 

4631 

4633 

4634    4636 

4637 

1 
2 
3 
4 
5 
6 
7 
8 
9 

4639 
4654 
4669 

4683 
4698 
4713 
4728 
4742 
4757 

4640 

4655 
4670 

4685 
4700 
47H 
4729 

4744 
4758 

4642 

4657 
4672 

4686 
4701 
4716 

473° 
4745 
4760 

4643 
4658 

4673 
4688 
4703 
4717 
4732 
4747 

4660 
4675 
4689 
4704 

4733 
4748 
4763 

4646 
4661 
4676 
4691 
4706 
4720 

4735 
4749 
4764 

4648 
4663 
4678 
4692 
4707 
4722 

4736 
4751 
4765 

4649 
4664 
4679 
4694 
4709 
4723 
4738 
4752 
4767 

4651 
4666 
4681 

4695 
4710 

4725 
4739 
4754 
4768 

4652 
4667 
4682 
4697 
4711 
4726 

4755 
4770 

4784 

i  300 

477i 

4773 

4774 

4776 

4777  1  4778 

4780 

478i 

4783 

i 

s   h  m   s 

II      O    1     II 

250  =  o  4  10 
260  =  o  4  20 
270  =  o  4  30 
280  —  o  4  40 
290  =  o  4  50 
300  =.05   o 

8   h  in   s 

II         Q         1        II 

2500  —o  41  40 
2600  ;=  o  43  20 
2700  =  o  45   o 
2800  —  o  46  40 

2900  =  0  48   20 

3000  =  o  50   o 

10 


TABLE  VI. 


Logarithms  of  Numbers  and  Small  Ares. 

No. 

O 

1 
4773 

2 

3 

4 

5 

£ 

7 

§ 

9 

47S4 

300 

477i 

4774    4776 

4777  . 

4778 

4780  |  4781 

4783 

1 
2 
3 
4 
5 
6 
7 
8 
9 

4786 
4800 
4814 
4829 
4843 
4857 
4871 
4886 
4900 

4787  i  4789 
4802  i  4803 
4816  j  4817 

4830  !  4832 

4844  i  4846 
4859  !  486o 

4873  i  4874 
4887  4888 
4901   4902 

4790 
4804 
4819 

4833 
4847 
4861 

4876 
4890 
4904 

479i 
4806 
4820 

4834 
4849 
4863 

4877 
4891 

4905 

4793 
4807 
4822 
4836 
4850 
4864 
4878 

4893 
4907 

4794 
4809 
4823 

4837 
4852 
4866 

4880 
4894 
4908 

4922 

4796 
4810 
4824 

4839 
4853 
4867 

4881 

4895 
4909 

4797 

4812 
4826 
4840 

4854 
4869 

4883 
4897 
4911 

4799 
4813 
4827  , 

4842  ! 
4856 
4870 
4884 
4898 

4912 

310 

~T 

2 
3 
4 

5 

6 

y 

§ 

9 

4914 

4915  |  4916 

4918- 

4919 

4921 

4923 

4925 

4926 

4928 
4942 
4955 
4969 
4983 
4997 
5011 
5024 
5038 

4929 

4943 
4957 

4971 
4984 
4998- 
5012 
5026 
5039 

493° 
4944 
4958 
4972 
4986 
5000 

5OI3 

5027 
5041 

4932 
4946 
4960 

4973 
4987 
5001 

5°i5 
5028 
5042 

4933 
4947 
4961 

4975 
4989 
5002 
5016 
5030 
5°43 

4935 
4949 
4962 

4976 
4990 
5004 

5OI7 
503i 
5°45 

4936 
4950 
4964 

4978 
4991 

5005 
5OI9 
5032 
5046 

4937 
4951 
4965 
4979 
4993 
5006 

5020 
5°34 
5°47 

4939 
4953 
4967 

4980 

4994 
5008 

5022 
5°35 
5°49 

4940 

4954 
4968 

4982 
4996 
5009 

5023 
5°37 
5°5° 

320 

1 

2 

3 
4 

5 
6 

7 
§ 
9 

5052 

5053 

5054 

5056 

5057 

5058 

5060 

5061 

5°75 
5088 
5101 

5"5 

5128 

5Hi 

5i55 
5168 
5181 

5062 

5064 

5065 

5°79 
5092 

5io5 
5"9 
5132 

5H5 
5159 

5'72 

5066 
5080 
5093 

5I07 
5120 

5134 

5H7 
5160 

5173 

5068 
5081 
5095 
5108 
5122 
5i35 
5H8 
5161 

5i75 

5069 
5083 
5096 

5109 
5123 
5136 

5H9 
5^3 
5J76 

5070 
5084 
5°97 

5"i 
5!24 
5138 

5T 
5l64 

5*77 

5072 
5085 
5°99 
5112 
5126 
5!39 
5152 
5i65 
5r79 

5073 
5087 
5100 

5"3 
5I27 
5  HO 

5153 
5i67 
5180 

ias 

5I03 
5116 
5*3° 
SMS 

5156 
5169 
5183 

5077 
5Q9i 
5104 
5118 
5131 
5*44 
5157 
51/1 
5184 

330 

5185 

5186 

5188 

5189 

5190 

5!92 

5193 

5i94 

5190 

319? 

5210 
5223 
5236 

5249 
5262 

5275 
5288 
53°i 
53H 

1 

2 
3 
4 
5 
6 
7 
8 
9 

5198 
5211 

5224 

5237 
5250 
5263 

5276 
5289 

5302 

5200 

5213 
5226 

5239 
5252 
5265 
5278 
5290 
53°3 

5201 

5214 
5227 

5240 

5253 
5266 

5279 
5292 
53°5 

5202 

5215 
5228 

5241 

5254 
5267 

5280 

5293 
5306 

5204 
5217 

5230 

5243 
5256 
5269 

5281 
5294 
5307 

5205 
5218 
5231 

5244 
5257 
5270 

5283 
5296 
5308 

5206    5207 
5219    5221 
5232    5234 

5245    5247 
5258    5260 
5271    5272 
5284  :  5285 
5297    5298 
53io    53" 

5209 
5222 
5235 
5248 
5261 
5274 
5287 
5299 
53M 

5325 

5338 
535° 
5363 

53g 
S35* 

54oi 

54i3 
5420 

5438 

340 

5315 

53i6 

5317 

53*9 

5320 

532i    5322 

5324 

5326 

1 
2 
3 
4 
5 
6 
7 
8 
9 

5328 
5340 

5353 
5366 
5378 
539i 

5403 
5416 
5428 

5329 
5342 
5354 
5367 
5379 
5392 

5405 
5417 
.  5430 

533° 
5343 
5355 
5368 
538i 
5393 
5406 
54i8 
543i 

533i 

5344 

5357 

5369 
5382 
5395 

5407 
5420 
5432 

5333 

5345 
5358 

537i 
5383 
5396 
5408 
5421 
5433 

5334 

5347 
5359 
5372 
5384 
5397 

54io 

5422 
5434 

5335 
5348 
536i 

5373 
5386 
5398 
54" 
5423 
,5436 

5336 
5349 
5362 

5374 
5387 
5400 

5412 
5425 
5437 

5339 

5352 
5364 
5377 
5390 
5402 

54i5 
5427 
5439 

350 

5441 

5442 

5443 

5444 

5446 

5447 

5448 

5449 

545i 

5452 

8   h  m   s                         s   h  m 

II     0    1     II                                               II     0     1 

300  =  05   o                       3000  =  o  50 
310  =  0  5  10                      3100  =  0  51 
320  =  o  5  20                      3200  =  0  53 

33°  =  o  5  3°                .      33°°  =  °  55 
340  =  o  5  40                      3400  =  o  56 
350  =  o  5  50                      3500  =  o  58 

s 
o 

40 

20 
o 

40 

20 

TABLE  VI. 


11 


Logarithms  of  Numbers  and  Small  Ares. 

No. 

0 

1 

2 

.3 

4 

5 

6 

7 

S 

9 

350 

544i 

5442 

5443  !  5444 

5446 

5447 

5448 

5449 

545i 

5452 

1 

2 

3 
4 
5 
6 
7 
8 
9 

5453 
5465 
5478 
5490 
5502 
55i5 
5527 
5539 
555i 

5454 
5467 
5479 
549i 
5504 
55i6 

5528 
5540 
5552 

5456 
5468. 
548o 
5492 
5505 
5517 
5529 
554i 
5553 

5457 
5469 
548i 

5494 
55o6 
55i8 

5530 

5542 
5555 

5458 
5470 
5483 
5495 
55°7 
5519 
5532 
5544 
5556 

5459 

5472 
5484 
5496 
5508 
552i 
5533 
5545 
5557 

5460 
5473 

5485 

5497 
55io 
5522 

5534 
5546 
5558 

5462 

5474 
5486 

5499 
55" 

5523 
5535 
5547 
5559 

5463 
5475 
5488 

55oo 
5512 
5524 
5536 
5549 
556i 

5464 
5477 
5489 
5501 
55i3 
5525 
5538 
5550 
5562 

36O 

5563 

5564 

5565    55°7 

5568 

5.569 

5570 

557i 

5573 

5574 

1 
2 
3 
4 
5 
6 
7 
8 
9 

5575 
5587 
5599 
5611 
5623 
5635 

5647 
5658 
5670 

5576 
5588 
5600 

5612 
5624 
5636 
5648 
5660 
5671 

5577 
5589 
5601 

5613 
5625 
5637 

5649 
5661 

5673 

5579 
559i 
5603 

5615 
5627 

5638 

5650 
5662 

5674 

558o 
5592 
5604 

5616 
5628 
5640 

5651 
5663 
5675 

558i 
5593 
5605 

5617 
5629 

5641 

5653 
5664 
5676 

5582 

5594 
5606 

5618 
5630 
5642 

5654 
5666 

5677 

5583 
5595 
5607 

56l9 
5631 
5643 

5655 
5667 
5678 

5585 
5597 
5609 

5621 
5632 
5644 

5656 
5668 
5680 

5586 
5598 
5610 

5622 
5634 
5645 

5657 
5669 
5681 

370 

5682 

5683 

5684 

5686  !  5687 

5688 

5689 

5690 

5691 

5693 

1 

2 

3 
4 

5 
6 

7 
§ 
9 

5694 
5705 
5717 
5729 
5740 
5752 
5763 
5775 
'  5786 

5695 
5707 
57i8 

5730 
574i 

5753 

5765 
5776 
5788 

5696 
57o8 
5719 
573i 

5743 
5754 
5766 
5777 
5789 

5697 
5709 
572i 

5732 
5744 
5755 
5767 
5778 
5790 

5698 
57io 
5722 

5733 
5745 
5757 
5768 
578o 
579i 

57°° 
57" 

5723 

5735 
5746 
5758 

5769 
578i 
5792 

57oi 
5712 
5724 
5736 
5747 
5759 
5770 
5782 
5793 

5702 
57*4 

5725 

5737 
5748 
5760 

577i 
5783 
5794 

57°3 
5715 
5726 

5738 
5750 
576i 

5773 
5784 
5796 

5704 
57i6 
5728 

5739 
575i 
5762 

5774 
5785 
5797 

380 

5798 

5799 

5800 

5801 

5802 

5804 

5805 

5806 

5807 

5808 

1 

2 
3 
4 

5 
6 
7 

8 
9 

5809 

5821 
5832 

5843 
5855 
5866 

5877 
5888 
5900 

5810 
5822 
5833 
5844 
5856 
5867 
5878 
5889 
590i 

5812 
5823 
5834 
5846 

5857 
5868 

5879 
5891 
5902 

5813 

5824 
5835 
5847 
5858 
5869 

5880 
5892 
5903 

58H 
5825 
5837 
5848 

5859 
5870 

5882 
5893 
5904 

5815 
5826 

5838 

5849 
5860 

5871 
5883 
5894 
5905 

5816 
5827 
5839 
5850 
5861 

5873 
5884 

5895 
5906 

5817 
5829 
5840 

5851 
5863 
5874 

5885 
5896 

5907 

5818 
5830 
5841 
5852 
5864 
5875 
5886 

5897 
5908 

5820 
5831 
5842 

5853 
5865 
5876 

5887 
5898 
59io 

39O 

59" 

5912 

5913 

59M    5915 

59i6 

5917 

5918  |  5920 

5921 

1 
2 
3 
4 
5 
6 
7 
8 
9 

5922 
5933 
5944 

5955 
5966 

5977 
5988 

5999 
6010 

5923 

5934 
5945 
5956 
5967 
5978 

5989 
6000 
60  1  1 

5924    5925    5926 
5935    5936    5937 
5946    5947    5948 

5957  :  5958  i  5959 
5968    5969    5970 

5979  :  598o    5981 
5990  !  5991    5992 

6OOI    :    6OO2       6OO3 
6012    I    6013       6014 

5927 
5938 
5949 
5960 

597i 
5982 

5993 
6004 
6015 

5928 
5940 
595i 
5962 
5973 
5984 

5994 
6005 
6016 

593° 
5941 
5952 
5963 
5974 
5985 
5996 
6006 
6017 

593  i 
5942 
5953 
5964 

§s 

H 

6018 

5932 
5943 
5954 
5965 
5976 
5987 
5998 
6009 
6020 

400 

6021    6022 

6023       6024       6025 

6026 

6027 

6028 

6029 

6030 

«   /i 

"     0 

350  =  0 

360  =  o 
-?7o  =  o 
380  =  0 
390  =  o 

400  =  o 

m   x                          a   h  m   s 

in                          n   &   i   it 

S   50                                3500  =  0   58   20 

6   o                       3600  =i   oo 
6  10                       37°°  —  i   i  40 
6  20                       3800  =i   3  20 
6  30                       3900  —  i   5   o 
6  40                       4000  =i   6  40 

12 


TABLE  VI. 


Logarithms  of  Numbers  and 

Small  Arcs. 

No. 

0 

1 

2 

3 

4 

5  . 

6 

7 

8 

9 

400 

6021 

6022    6023 

6024 

6025 

6026 

6027 

6028 

6029 

6030 

1 
2 
3 
4 
A 
6 
7 
8 
9 

6031 
6042 
6053 
6064 
6075 
6085 
6096 
6107 
6117 

6033 
6043 
6054 
6065 
6076 
6086 
6097 
6108 
6118 

6034 
6044 
6055 
6066 
6077 
6087 
6098 
6109 
6119 

SI 

6056 
6067 
6078 
6088 

6099 
6110 
6120 

6036 
6047 
6057 
6068 
6079 
6090 
6100 
6m 
6121 

6037 
6048 
6058 
6069 
6080 
6091 
6101 

6lI2 

6123 

6038 
6049 
6060 
6070 
6081 
6092 
6102 
6113 
6124 

6134 

6039 
6050 
6061 
6071 
6082 
6093 
6103 
6114 
6125 

6040 
6051 
6062 
6072 
6083 
6094 
6104 
6115 
6126 

6041 
6052 
6063 
6073 
6084 
6095 
6106 
6116 
6127 

SCO 

6128 

6129 

6130 

6131 

6132 

6l33 

6135 

6146 
6156 
6167 
6177 
6188 
6198 
6209 
6219 
6229 

6136 

«"37 

1 

2 
3 
4 

5 
6 

7 
8 
9 

6138 
6149 
6160 
6170 
6180 
6191 
6201 
6212 
6222 

6139 
6150 
6161 
6171 
6182 
6192 
6202 
6213 
6223 

6141 
6151 
6162 
6172 
6183 
6193 
6203 
6214 
6224 

6142 
6152 
6163 

6i73 
6184 
6194 
6204 
6215 
6225 

6i43 
6i53 
6164 

6174 
6185 
6i95 
6206 
6216 
6226 

6237 

6144 

6i54 
6165 

6i75 
6186 
6196 
6207 
6217 
6227 

6145 
6i55 
6166 

6176 
6187 
6197 
6208 
6218 
6228 

6147 

6i57 
6168 

6178 
6189 
6199 
6210 
6220 
6230 

6148 
6158 
6169 
6179 
6190 
6200 
6211 
6221 
6231 

6242 

I  '2  > 

6232 

6234 

6235 

6236 

6238 

6239 

6240 

6241 

1 
2 
3 
4 
5 
6 
7 
8 
9 

6243 
6$ 

6274 
6284 
6294 
6304 
63H 
6325 

6244 
6254 
6264 

till 

6295 
6305 

^l 
6326 

6245 

6255 
6265 

6276 
6286 
6296 
6306 
6316 
6327 

6246 
6256 
6266 
6277 
6287 
6297 
6307 

6$ 

6247 

6268 
6278 
6288 
6298 
6308 
6318 
6329 

6248 
6258 
6269 
6279 
6289 
6299 
6309 
6320 
633° 

6249 
6259 
6270 
6280 
6290 
6300 
6310 
6321 
633J 

6250 
6260 
6271 
6281 
6291 
6301 
6311 
6322 
6332 

6251 
6261 
6272 
6282 
6292 
6302 
6312 
6323 
6333 

6252 
6262 
6273 
6283 
6293 
6303 

6313 
6324 

6334 

i  430 

6335 

6336 

6337 

6338 

6339 

6340 

6341 

6342 

6343 

6344 

1 
tt 
3 
4 
5 
6 
7 
8 
9 

6345 
6355 
6365 

6375 
6385 
6395 
6405 

6415 
6425 

6346 
6$ 

6$ 
6396 
6406 
6416 
6426 

6347 
6357 
6367 

6377 
6387 
6397 
6407 
6417 
6427 

6348 
6358 
6368 
6378 
6388 
6398 
6408 
6418 
6428 

6349 
6359 
6369 

6379 
6389 

6399 
6409 
6419 
6429 

6350 
6360 
6370 
6380 
6390 
6400 
6410 
6420 
6430 

635i  ' 
6361 

637i 
6381 
6391 
6401 
6411 
6421 
643  * 

6352 
6362 
6372 
6382 
6392 
6402 
6412 
6422 
6432 

6353 
6363 
6373 
6383 
6393 
6403 

6413 
6423 

6433 

6354 
6364 

6374 
6384 

6394 
6404 

6414 
6424 
6434 

440 

6435 

6436 

6437 

6437 

6438 

6439 

6440 

6441 

6442 

6443 

1 
2 
3 

f 

6 

7 
§ 
9 

6444 

6454 
6464 

6474 
6484 
6493 
6503 
6513 
6522 

6445 
6455 
6465 

6475 
6485 
6494 
6504 

6514 
6523 

6446 
6456 
6466 
6476 
6486 
6495 
6505 
6515 
6524 

6447 

6457 
6467 

6477 
6487 
6496 
6506 
6516 
6525 

6448 
6458 
6468 
6478 
6488 
6497 
6507 

6517 
6526 

6449 

6459 
6469 

6479 
6488 
6498 
6508 
6518 
6527 

6450 
6460 
6470 
6480 
6489 
6499 
6509 

6519 
6528 

6451 
6461 
6471 
6481 
6490 
6500 
6510 
6520 
6529 

6452 
6462 
6472 
6482 
6491 
6501 
6511 
6521 
6530 

6453 
6463 

6473 
6483 
6492 
6502 
6512 
6522 
653! 

;  45O 

6532 

6533 

6534  !  6535 

6536 

6537  |  6538 

6539 

6540 

6541 

s   h  m  s                           s   h  m   s 

II       Q      1       II                                                                  II      0       1      It 

400  =  o  6  40                       4000  =    6  40 
410  =  o  6  50                       4100  =    S  20 
420  =  07   o                       4200  =   10   o 
430  =  0  7  10                       43°°=   IJ  40 
440  =  o  7  20                       4400  —   13  20 
45°  =  °  7  3°                      45°°  =   *5   c 

TABLE  VI. 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 
450 

1 
g 

8 
4 
5 
6 

7 

8 
9 

o    i    a    s 

4 

567 

8 

6540 

9 

<>532 

6542 

6551 
6561 

6571 
6580 
6590 

6599 
6609 
6618 

6533 

6534 

6535 

6545 
6554 
6564 

6573 
6583 
6593 
6602 
6612 
6621 

6536 

6537 

6538    6539 

6541 

6543 

6552 
6562 

6572 
6581 
6591 
6600 
6610 
6619 

6544 
6553 
6563 
6572 
6582 
6592 
6601 
6611 
6620 

6546 

6555 
6565 

6574 
6584 

6593 
6603 
6612 
6622 

6547 
6556 
6566 

6575 
6585 
6594 
6604 
6613 
6623 

6548 
6557 
6567 

6576 
6586 
6595 
6605 
6614 
6624 

6549 
6558 
6568 

6577 
6587 
6596 
6606 
6615 
6625 

6549 
6559 
6569 
6578 
6588 
6597 
6607 
6616 
6626 

6550 
6560 
6570 

6579 
6589 
6598 
6608 
6617 
6627 

460 

6628 

6629 

6629 

6630 

6631 

6632 

6633 

6634 

6635 

6636 

1 

2 
3 
4 

5 
0 
7 

8 
9 

6637 
6646 
6656 
6665 

6684 

6693 
6702 
6712 

6638 
6647 
6657 

6666 

6675 
6685 

6694 
6703 
6713 

6639 
6648 
6658 
6667 
6676 
6686 

6695 
6704 
6714 

6640 
6649 
6659 
6668 
6677 
6687 
6696 
6705 
6715 

6641 
6650 
6660 
6669 
6678 
6688 
6697 
6706 
6715 

6642 
6651 
6661 
6670 

6679 
6689 

6698 
6707 
6716 

6643 
6652 
6661 
6671 
6680 
6689 
6699 
6708 
6717 

6644 
6653 
6662 

6672 
6681 
6690 
6700 
6709 
6718 

6645 
6654 
6663 

6673 
6682 
6691 
6701 
6710 
6719 

6645 
6655 
6664 

6674 
6683 
6692 
6702 
6711 
6720 

470 

6721 

6722 

6723 

6724 

6725 

6726 

6727  |  6727 

6728 

6729 

1 
2 
3 

4 

5 

6 
7 

§ 
9 

6730 
6739 
6/49 
6758 
6767 
6776 
6785 

6794 
6803 

6731 
6740 
6750 

6759 
6768 

6777 
6786 

6795 
6804 

6732 
6741 
6750 
6760 
6769 
6778 
6787 

8g 

6733 
6742 
6751 
6761 
6770 
6779 
6788 

&i 

6734 
6743 
6752 

6761 
6771 
6780 
6789 
6798 
6807 

6735 
6744 
6753 
6762 
6772 
6781 
6790 

6799 
6808 

6736 
6745 
6754 
6763 
6772 
6782 
6791 
6800 
6809 

6737 
6746 

6755 
6764 
6773 
•6783 
6792 
6801 
6810 

6738 

6747 
6756 

6765 
6774 

6783 
6792 
6802 
6811 

6739 
6748 
6757 
6766 
6775 
6784 

6793 
6802 
6812 

480 

6812 

6813 

6814 

6815 

6816 

6817 

6818 

6819 

6820 

6821 

1 

2 
3 

! 

7 
i   8 
9 

6821 
6830 
6839 
6848 
6857 
6866 

6875 
6884 
6893 

6822 
6831 
6840 
6849 
6858 
6867 
6876 
6885 
6894 

6823 
6832 
6841 
6850 
6859 

6868 
6877 
6886 
6895 

6824 

6833 
6842 

6851 
6860 
6869 
6878 
6887 
6896 

6825 
6834 
6843 
6852 
6861 
6870 

6879 
6888 

6897 

6826 

6835 
6844 

'6853 
6862 
6871 
6880 
6889 
6898 

6827 
6836 
6845 
6854 
6863 
6872 
6881 
6890 
6898 

6828 
6837 
6846 

3g 

6873 
6882 
6890 
6899 

6829 
6838 
6847 
6856 
6865 
6874 
6882 
6891 
6900 

6830 
6839 
6848 
6857 
6865 
6874 
6883 
6892 
6901 

49O 

6902 

6903 

6904 

6905 

6906 

6906 

6907 

6908 

6909 

"  6918 
6927 
6936 
6944 
^953 
6962 

6971 
6979 
6988 

6910 

1 

3 

f 

6 

7 
8 
9 

6911 
6920 
6928 

6937 
6946 

6955 
6964 
6972 
6981 

6912 
6921 
6929 
6938 
6947 
6956 
6964 

6973 
6982' 

6913 
6921 
6930 

6939 
6948 

6957 
6965 
6974 
6983 

6913 
6922 
6931 
6940 
6949 
6957 
6966 

6975 
6984 

6914 

6923 
6932 

6941 
6950 
6958 

6967 
6976 
6985 

6915 
6924 

6933 
6942 
6950 
6959 
6968 

6977 
6985 

6916 
6925 
6934 
6943 
6951 
6960 

6969 
6978 
6986 

6917 
6926 

6935 
6943 
6952 
6961 

6970 
6978 
6987 

6919 

6928 
6936 

6945 
6954 
6963 

6971 
6980 
6989 

500 

6990 

6991 

6991    6992 

6993 

6994 

6995 

6996 

6997 

•  6998 

s   It  m   s                           s   k  m   s 

II      0     1     II                                                   II     o      1      II 

450  —  0  7  30                       4500  =115   o 
460  =  o  7  40                       4600  =i  16  40 
470  =  o  7  50                       4700  =i  18  20 
480  =  08   o                       4800  =  i  20   o 

490  r=  O   8   IO                                  49OO=i  I   21   40 

500  =  o  8  20                       5000  =  i  23  20 

14 


TABLE  VI. 


1  No. 

Logarithm*  of  Number*  and  Small  Arcs. 

8 

o  i  i 

234 

5 

6 

7 

9 

5OO 

6990 

6991 

6991 

6992 

7001 
7010 
7018 
7027 

7035 
7044 

7053 
7061 
7070 

6993 

7002 
7011 
7019 
7028 
7036 
7045 

7°54 
7062  i 
7071 

6994 

6995 

6996 

6997  ! 

6998 

7006 

7015 
7023 

7032 
7041 
7049 

7058 
7066 

7075 

1 
2 
3 
4 
5 
6 
7 
8 
9 

6998 
7007 
7016 
7024 

7033 
7042 

7050 

7059 
7067 

6999 
7008 
7017 
7025 

7034 
7042 

7051 
7059 
7008 

7000 
7009 
7017 
7026 
7035 
7043 
7052 
7060 
7069 

7003 
7011 
7020 
7029 

7037 
7046 

7054 
7063 
7071 

7004 
7012 
7021 
7029 
7038 
7047 

7055 
7064 
7072 

7004 

7013 
7022 

7030 

7039 
7048 

7056 
7065 
7<>73 

7005  i 
7014  : 
7023 

703  I 
7040 
7048 

7057 
7074 

510 

7076 

7077 

7077 

7078  i  7079 

7080 

7081 

7082 

7083  j   7083 

1 
2 
3 
4 
5 
6 
7 

9 

7084 

7093 
7101 

7110 
7118 
7127 

7135 
7H3 
7152 

7085 
7094 
7102 
7110 
7119 
7127 

7136 
7144 
7153 

7086 
7094 
7103 
7111 
7120 
7128 

7137 

7H5 
7153 

7162 

7087 

7095 
7104 

7112 
7121 
7129 

7137 
7146 

7154 

7088 
7096 
7105 

7121 
7130 

7138 
7147 
7155 

7088 
7097 

7114 

7122 
7131 

7139 
7H7 

7106 

7U5 
7123 
7132 
7140 
7148 
7157 

7090 
7099 
7107 
7116 
7124 
7132 
7141 
7H9 
7158 

7091 
7099 
7108  ; 
7116 
7125 
7133 

7142  ; 
7150 
7158 

7092 
7100 
7109 
7117 
7126 
7134 
7142 
7151 

5  2O 

7160 

7161 

7163 

7163 

7164 

7165 

7166 

7167  i 

7168 

1 

3 
4 

5 
6 

7 
8 
9 

7168 

7177 
7185 

7193 
7202 
7210 
7218 
7226 
7235 

7194 
7202 
7211 

7219 

7227 

7235 

7170 
7178 
7187 

7195 
7203 
.  7212 
7220 
7228 
7236 

7171 

71/9 
7188 

7196 
7204 
7212 
7221 
7229 
7237 

7172 
7180 
7188 
7197 
7205 
7213 
7221 
7230 
7238 

7173 
7181 
7189 

7197 
7206 
7214 
7222 
7230 
7239 

7173 
7182 
7190 
7198 
7207 
7215 
7223 
7231 
7239 

7174 
7183 
7191 

7199 
7207 
7216 
7224 
7232 
7240 

7175 

7183  ; 
7192 

7200 
7208 
7216 

7225 

7233 
7241 

7176 
7184 
7192 
7201 
7209 
7217 
7226 

7234 
7242 

530 

7243 

7244 

7244 

7245 

7246 

7247 

7248 

7248 

7249 

7257 
7266 

7274 
7282 
7290 
7298 
7306 

73H 
7322 

7250 

1 
2 
3 
4 
5 
6 
7 
8 
9 

7251 
7259 
7267 

7275 
7284 
7292 
7300 
7308 

7252 
7260 
7268 
7276 
7284 
7292 

7301 
73°9 
7V7 

7253 
7261 
7269 

7277 
7285 
7293 
7301 
73°9 

7253 

7262 
7270 
7278 
7286 
7294 
7302 

7254 
7262 
7271 
7279 
7287 
7295 
7303 
73" 
73  19 

7255 
7263 
7271 
7279 
7288 
7296 

73°4 
7312 
7320 

7256 
7264 
7272 
7280 
7288 
7297 
7305 
7313 
7321 

7257 
7265 

7273 
7281 
7289 
7297 
73°5 

7322 

7258 
7266 

7275 
7283 
7291 
7299 
73°7 

7323 

540 

7324 

7325 

7326 

7326 

7327 

7328 

7329 

733° 

7330     7331 

1 
2 
3 
4 

6 

7 
8 
9 

7332 
7340 
7348 

7356 
7364 
7372 
7380 
7388 
7396 

7333 
734i 
7349 
7357 
7365 
7373 

7389 
7397 

7334 
7342 
7350 

7366 
7374 
738i 
7389 
7397 

7334 
7342 
7350 
7358 
7366 

7374 
7382 
7390 
7398 

7335 
7343 
7351 
7359 
7367 
7375 
7383 

7399 

7336 
7344 
7352 
7360 
7368 
7376 

7384 
7392 
7400 

7337 
7345 
7353 
7361 
7369 
7377 

'7385 
7393 
7400 

7338 
7346 
7354 
7362 
7370 
7377 
7385 
7393 
7401 

7338 
7340 
7354 
7362 

7386 
7394 
7402 

7339 
7347 
7355 
7363 
737i 
7379 
7387 
7395 
7403 

55O 

7404 

7404 

7405 

7406 

7407 

7408 

7408 

7409 

7410 

7411 

s   h  m   s                           s   h  m   s 

1.     0    /     //                                               H     0     1     'I 

500  =  0   8   20                                5000  =1   23   20 

510-0  8  30                      5100  =  i  25   o 
520  =  o  8  40                       5200  =i  26  40 
530  —  o  8  50                       5300  =i  28  20 
540  =  09   o                       540°  =13°   ° 
550  =  o  9  jo                       5500  =  i  31  40 

TABLE  VI. 


15 


No. 

L.O; 
0             1 

^aritlim*  of  Numbers  and  Small  Ares. 

2 

3 

4 

5 

6 

~74o8 

7 

8 

9 

55O 

7404 
7412 

74^9 
7427 

7435 
7443 
745i 

7459 
7466 

7474 

7404 

7405 

74i3 
7421 
7429 

7437 
7444 
7452 
7460 
7468 
7476 

7406 

7407 

7408 

7409 

7410 

74" 

1 

2 
3 
4 
5 
6 

7 
8 
9 

7412 
7420 
7428 

7436 
7444 
7452 

7459 

7407 

7475 
7483 

74H 

7422 

743° 

7437 
7445 
7453 
7461 
7469 
7476 

7415 
7423 
743° 

7438 
7446 

7454 
7462 
7469 

7477 

7415 
7423 
743  r 
7439 
7447 
7455 
7462 
747° 
7478 

7416 
7424 
7432 
7440 
7448 

7455 
7463 
747i 
7479 

7417 
7425 
7433 
744i 
744s 
7456 
7464 
7472 
7480 

7418 
7426 
7434 
744i 
7449 
7457 
7465 
7473 
7480 

7419 
7426 

7434 
7442 
745g 
7458 
7466 

7473 
748i 

560 

7482 

7483         7484         7485 

7486         7487 

7487 

7488           7489 

1 

2 

3 

4 

5 
6 

7 
8 
9 

7490 
7497 
75°5 

75*3 
7520 
7528 

7536 
7543 
755i 

749° 
7498 
7506 

75  H 
752i 
7529 

7537 
7544 
7552 

75  60 

7491          7492 
7499         7500 
75°7         75°7 
7514    !.     7515 

7522         7523 
753°    f    753° 
7537    ,     7538 
7545         7546 
7553    !     7553 

7493 
7500 
7508 
75i6 

7524 
753i 
7539 

7547 
7554 

7494 
7501 
7509 

7517 
7524 
7532 
7540 
7547 
7555 

7494 
7502 
75io 

7517 

7525 
7533 
7540 
7548 
7556 

7495 
7503 
75  10 
75i8 
7526 
7534 
754i 
7549 
7556 

7496 
75°4 
75" 
7519 

7527 
7534 
7542 
7550 
7557 

7497 
7504 
7512 

7520 

7527 
7535 

7543 
7550 

7558 

570 

v'.-l 
2 
3 
4 
5 
6 
7 
* 
9 

7559 

7560         7561 

7562 

7563 

7563         7564 

7565 

7566 

7566 

7574 
7582 

7589 
7597 
7604 

7612 
7619 
7627 

7567 
7575 
7582 

7590 
7597 
7605 

7613 
7620 
7628 

7568 

75|S 

7s°3 

759i 
7598 
7606 

7613 
7621 

7628 

7569 
7576 
7584 
759i 
7599 

7614 
7622 
7629 

7569 
7577 
7585 
7592 
7600 
7607 

7615 
7622 
7630 

7570 
7578 
7585 

7593 
7600 
7608 
7616 
7623 
763! 

757i 
7579 
7586 

7594 
7601 
7609 
7616 
7624 
7631 

7572 
7579 
7587 

7594 
7602 
7610 
7617 
7625 
7632 

7572 
758o 
7588 

7595 
7603 
7610 
7618 
7625 
7633 

7573 
758i 
7588 

7596 
7603 
7611 
7619 
7626 
7634 

580 

7634 

7635 

7636    j     7637 

7637 

7638 

7639 

7640 

7640 

7641 

1 

2 
3 
4 

5 
6 

7 
8 
9 

7642 
7649 
7657 
7664 
7672 
7679 
7686 
7694 
7701 

7643   ' 
7650 

7657 
7665 
7672 
7680 

7687 

7695 
7702 

7643 
7651 
7658 

7666 

7673 
7680 

7688 
7695 
7703 

7644 
7651 
7659 
7666 

7674 
7681 

7689 
7696 
7703 

7645 

7652 
7660 
7667 

7675 
7682 

7689 
7697 
7704 

7646 

7653 
7660 

7668 

7675 
7683 
7690 
7697 
7705 

7646 

7654 
7661 

7669 
7676 
7683 
7691 
7698 
7706 

7647 
7654 
7662 

7669 
7677 
7684 
7692 
7699 
7706 

7648 

7655 
7663 

7670 
7677 
7685 
7692 
7700 
7707 

7648 
7656 
7663 
7671 
7678 
7686 

7693 
7700 
7708 

59O 

7709 

7709 

7710 

77" 

77" 

7712 

7713 

77H 

77H 

7715 

1 

2 
3 
4 
5 
6 

8 
9 

7716 

7723 
773i 
7738 
7745 
7752 
7760 
7767 
7774 

7717 
7724 
773i 

7739 
7746 

7753 
7760 
7768 
7775 

7717 
7725 
7732 
7739 
7747 
7754 
7761 
7768 
7776 

7718 
7725 
7733 
7740 
7747 
7755 
7762 
7769 
7776 

7719 
7726 
7733 

774i 
7748 
7755 

7763 
7770 

7777 

7720 

7727 
7734 
7742 
7749 
7756 

7763 
7771 

7778 

7720 
7728 
7735 
7742 
775° 
7757 
7764 
7771 
7779 

7721 
7728 
7736 

7743 
7750 
7758 

7765 
7772. 
7779 

7722 
7729 
7736 
7744 
775i 
7758 
7766 

7773 
778o 

7722 
773° 
7737 
7744 
7752 
7759 
7766 
7774 
778i 

600 

7782 

7782 

7783 

7784 

7784 

7785 

7786 

7787 

7787 

7788 

s       h    m      s                                                                     8       h     m       $ 

II              O           1           II                                                                                                                              II              o              1            II 

550  =  o    9     10                                                         5500  =       31    40 
560  =  o    9    20                                                         5600  =       33    20 
570  =  o    9    30                                                       5700  =       35      o 

580  =r  O      9       4O                                                                                                58OO  =            36      4O 

590  =  o    9    50                                                         5900  =       38    20 
600  =  o  10      o                                                         6000  =       40      o 

16 


TABLE  VI. 


Logarithms  of*  Numbers  and  Small  Arcs. 

No. 

o 

1 

2 

«j 

t 

5 

6 

7 

8      9 

600 

7782 

7782 

7783 

7784 

7784 

7785 

7786 

7787 

7787      77KX  : 

1 
2 
3 
4 
5 
6 

7 
§ 
9 

7789 
7796 
7803 
7810 
7818 
7825 

7832 
7839 
7846 

7789 

7804 
7811 
7818 
7825 

7833 
784o 

7847 

7790 

7797 
7805 

7812 
7819 
7826 

7833 
7840 
7848 

7791 
7798 
7805 

7813 
7820 
7827 

7834 
7841 
7848 

7792 
7799 
7806 

7813 
7820 

7828 

7835 
7842 
7849 

7792 
7800 
7807 

7814 
7821 
7828 

7835 
7843 
7»5o 

7793 
7800 

7807 

7815 
7822 
7829 
7836 

7843 
7850 

7794 
7801 
7808 

7815 
7823 
7830 

7837 
7844 
7851 

7795 
7802 

7809 
7810 
7823 
7830 
7838 

7845 
7852 

7795 

7802 
78  10 

7817  ! 
7824 

7831 
7838 
7845 
7853 

610 

7853 

7854 

7855 

7855    7856 

7857 

7864 
7871 

7878 
7885 
7892 
7899 
7906 

79i3 
7920 

7858 

7858 

7859     786o  i 

1 

2 
3 
4 
5 

6 
7 

8 
9 

7860 
7868 
7875 
7882 
7889 
7896 

7903 
7910 

7917 

7861 
7868 
7875 
7882 
7889 
7897 
7904 
7911 
7918 

7862 
7869 
7876 

7883 
7890 
7897 
7904 

79  M 
7918 

7863 
7870 
7877 
7884 
7891 
7898 

7905 
7912 

7919 

7863 
7870 
7877 
7885 
7892 
7899 
7906 

7913 
7920 

7865 
7872 
7879 
7886 

7893 
7900 

7907 
79H 
7921 

7865 

7872 
7880 

7887 

7894 
7901 

7908 

7915 
7922 

7866 

7873 
7880 

7887 
7894 
7901 
7908 
7916 
7923 

7867  i 

7874  ! 
7881  i 

7888   ; 

7895  ! 
7902  ) 

7909 
7916  ! 

7923  ! 

620 

7924 

7925 

7925 

7926    7927 

7927 

7928 

7929 

7930    7930 

1 
2 
3 
4 
£ 
6 
7 
i 
9 

7931 
7938 
7945 
7952 
7959 
7966 

7973 
7980 

7987 

7932 
7939 
7946 

7953 
7959 
7966 

7973 
7980 

7987 

7932 
7939 
7946 

7953 
7960 
7967 

7974 
798i 
7988 

7933 
7940 

7947 

7954 
7961 
7968 

7975 
7982 
7989 

7934 
794i 
7948 

7955 
7962 
7969 

797  S 
7982 

7989 

7934 
794i 
7948 

7955 
7962 
7969 
7976 

7983 
7990 

7935 

7942 

7949 
7956 
7963 
7970 

7977 
7984 
7991 

7936 

7943 
795° 

7957 
7964 
7971 
7978 

7984 
7991 

7917 
7943 
795° 

7957 
7964 

7971 
7978 
7985 
7992 

7937  ; 
7944  i 
7951 
7958  i 
7965  ' 
7972  : 

7979 
7986  ! 

7993 

63O 

7993 

7994 

7995 

7995 

7996 

7997 

7998 

7998 

7999  |   8000 

1 
2 
3 
4 
5 
6 
7 
8 
9 

8000 
8007 
8014 
8021 
8028 
8035 
8041 
8048 
8055 

8001 
8008 
8015 
8022 
8028 
8035 
8042 
8049 
8056 

8002 
8009 
8015 
8022 
8029 
8036 

8043 
8050 
8056 

8002 
8009 
8016 
8023 
8030 
8037 
8043 
8050 
8057 

8003 
8010 
8017 
8024 
8030 
8037 
8044 
8051 
8058 

8004 
8011 
8017 
8024 
8031 
8038 

8045 
8052 
8058 

8004 
8011 
8018 
8025 
8032 
8039 

8045 
8052 
8059 

8005 
8012 
8019 
8026 
8033 
8039 
8046 

S32 

8006 
8013 
8020 
8026 

8033 
8040 

8047 

8054 
8060 

8006  i 
8013 
8020  i 
8027 

8034  ! 

8041  1 

8048  1 

8054  ! 
8061 

640 

8062 

8062  |  8063 

8064 

8065 

8065 

8066 

8067 

8067 

8068 

1 
2 
3 
4 
5 
6 
7 
9 
9 

8069 

8075 
8082 

8089 
8096 
8102 
8109 
8116 
8122 

8069 
80$ 
8083 
8090 
8096 
8103 
8110 
8116 
8123 

8070 

8077 
8083 

8090 

8097 
8104 

8110 
8117 
8124 

8130 

8071 

8077 
8084 

8091 
8098 
8104 
8111 
8118 
8124 

8071 
8078 
8085 
8092 
8098 
8105 
8112 
8118 
8125 

8072 
8079 
8085 

8092 
8099 
8106 
8112 
8119 
8126 

8073 
8079 
8086 
8093 
8100 
8106 

8113 
8120 
8126 

8073 
8080 
8087 

8094 
8100 
8107 
8114 
8120 
8127 

8074 
8081 
8088 
8094 
8101 
8108 
8114 
8121 
8128 

8075 
8081 
8088  i 

8095  1 
8102  i 
8108 

8115 
8122  i 
8128  1 

8135 

650 

8129 

8130 

8131    8132 

8132 

8i33 

8134 

8i34 

s   h  m   s                         s   h  in 

II       Q     1       II                                                                  II      O        1 

600  =  o  10   o                       6000  =•   40 
610  =  o  10  10                       6100  =   41 
620  =  o  10  20                       6200  =   43 
630  =  o  10  30                       6300  =   45 
640  =  o  10  40                       6400  =   46 
650  =  o  10  50                       6500  =   48 

8 

it 

0 
40 
2O 
O 

40 

20 

TABLE  VI. 


17 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

O 

1     2 

3 

4  • 

567 

8 

9 

650 

8129  | 

8129    8130 

8131 

8132 

8132    8133 

8134    8134 

8i35 

1 

2 
3 
4 
5 
6 

8 
9 

8136 
8142 
8149 
8156 
8162 
8169 
8176 
8182 
8189 

8136 

8i43 
8150 

S 

8176 
8183 
8190 

8137 

8144 
8150 

8157 
8164 
8170 
8177 

8190 

8138 
8144 
8151 
8158 
8164 
8171 
8178 
8184 
8191 

8138 

8i45 
8152 

8158 
8165 
8172 
8178 
8185 
8191 

8139  |  8140 
8146  i  8146 
8152  i  8153 
8159  :  8160 
8166  |  8166 
8172  1  8173 
8179  i  8180 
8186   8186 
8192   8193 

8140    8141 
8147    8148 
8154    8154 
8160    8161 
8167  :  8168 
8174  i  8174 
8180  ',  SiSi 
8187  !  8188 
8193    8194 

8142 
8148 

8155 
8162 

8168  ! 
8175  1 
8182 
8188 
8195 

660 

8i95 

8196 

8197 

8i97 

8198 

8199 

8199 

8200 

8201 

8201 

1 
2 
3 
4 
5 
6 

9 

8202 
8209 
8215 
8222 
8228 
8235 
8241 
8248 
8254 

8203 

8209 
8216 
8222 
8229 
8235 
8242 
8248 
8255 

8203 
8210 
8216 
8223 

8230 

8236 

8243 
8249 
8256 

8204 
8211 

8217 
8224 
8230 
8237 

8243 
8250 
8256 

8205 
8211 

8218 
8224 
8231 
8237 
8244 
8250 
8257 

8205 
8212 
8218 
8225 
8231 
8238 
8245 
8251 
8258 

8206 
8213 
8219 

8226 
8232 
8239 
8245 
8252 
8258 

8207 
8213 
8220 
8226 
8233 
8239 
8246 
8252 
8259 

8207 

8214 
8220 
8227 

8233 
8240 

8246 

8253 
8259 

8208 
8214 
8221 
8228  - 

8234 
8241 

8247 
8254 
8260 

67O 

8261 

8261   8262 

8263 

8263 

8264 

8265 

8265    8266 

8267 

1 
2 
3 
4 
5 
6 

9 

8267 

8274 
8280 

8287 

8293 
8299 
8306 
8312 
8319 

8268   8269 
8274  8275 

8281  '  8281 
8287   8288 

8294   8294 
8300  '  8301 

8307   8307 
8313  8314 
8319   8320 

8269 
8276 
8282 
8289 
8295 
8301 
8308 

8314 
8321 

8270 
8276 
8283 
8289 
8296 
•  8302 
8308 

8315 
8321 

8270  \  8271 
8277  '  8278 
8283  j  8284 
8290   8290 
8296  !  8297 
8303  :  8303 
8309    8310 
8316    8316 
8322    8323 

8272    8272 
8278  I  8279 

8285  ;  8285 
8291   8292 

8298   ;   8298 
8304     8305 
8310     8311 
8317   j   8317 
8323     8324 

8273 
8280 
8286 
8292 
8299 
8305 
8312 
8318 
8324 

680 

8325 

8326   8326 

8327 

8328 

8328 

8329 

8330 

8330 

8331 

£337 

8344 
8350 

8356 

8363 
8369 

8375 
8382 

1 
2 
3 
4 
5 
6 

9 

8332 
8338 
8344 
835i 
8357 
8363 
8370 
8376 
8382 

8332 

8338 

8345 

8351 
8358 
8364 

8370 

8377 

8383 

8333 
8339 
8345 
8352 
8358 

8365 
8371 
8377 
8383 

8333 
8340 
8346 

8352 
8359 
8365 
8371 
8378 
8384 

8334 
8340 

8347 
8353 
8359 
8366 

8372 
8378 
8385 

8335 
8341 
8347 

8354 
8360 
8366 

8373 
8379 
8385 

8335 
8342 
8348 

8354 
8361 

8367 

8373 
8380 
8386 

8336 
8342 
8349 

8355 
8361 
8368 

8374 
8380 

8387 

8337 
8343 
8349 
8356 
8362 
8368 

8375 
8381 
8387 

690 

8388 

8389 

8390 

8390 

8391 

8392 

8392 

8393 

8394 

8394 

1 

2 
3 
4 
5 
6 

8 
9 

8395 
8401 
8407 
8414 
8420 
8426 

8432 
8439 

8445 

8395 

8402 

8408 
8414 

8420 

8427 

8433 

8439 
8445 

8396 

8402 

8409 

8415 

8421 

8427 
8434 

8440 

8397 
8403 
8409 
8415 
8422 
8428 

8434 
8440 
8447 

8397 
8404 
8410 
8416 
8422 
8429 

8435 
8441 

8447 

8398 
8404 
8410 

8417 
8423 
8429 

8435 
8442 
8448 

8399 
8405 
8411 
8417 
8424 
8430 

8436 
8442 
8449 

8399 
8405 
84I2 
8418 

8424 
8430 

8437 
8443 
8449 

8400 
8406 
8412 
8419 
8425 
8431 

8437 
8444 
8450 

8400 
8407 
8413 
8419 
8425 
8432 
8438 
8444 
8450 

7OO 

8451 

8452 

8452 

8453 

8453 

8454 

8455 

8455 

8456 

8457 

s   h  m  s 

II      O    1     II 

650  =  o  10  50 
660  =  o  1  1   o 
670  =  011  10 

680  =  011   20 

690  =  o  1  1  30 
700  =011  40 

s 
n 

6500  = 
6600  = 
6700  = 
6800  = 
6900  - 
7000  = 

k  m  s 

0     /     // 

=   48   20 

r   50   o 
=  51  40 

=  53  20 

=  56  40 

18 


TABLE  VI. 


Logarithms  of  Numbers  and  Small  Ares. 

No. 

0 

1 

2 

3      1 

5     6 

7 

8455 

8 

9 

700 

8451 

8452 

8452 

8453  i  8453  - 

«454  : 

«455 

8461 
8467 
8473 

8479 
8486 
8492 
8498 
8504 
8510 

8456 

8457  i 

1 
? 
3 
4 
5 
6 
7 
8 
9 

8457 
8463 
8470  j 
8476 
8482 
8488 

8494 
8500 
8506 

8458 
8464 
8470 

8476 

8^3 
8489 

8495 
8501 

8507 

8458 
8465 
8471 

8477 
8483 
8489 

8495 
8502 
8508 

8459    8460 
8465    8466 
8471    8472 
8478    8478 
8484  ;  8484 
8490  !  8491 
8496  '  8496 
8^02    8503 
8508    8509 

8460 
8466 
8473 

8479  i 
8485 
8491 

8497 
8503 
8510 

8462 

8468 
8474 
8480 
8486  • 
8492 
8498 

8505 
8511 

8462 
8468 
8474 
8481 
8487 
8493 
8499 
8505 
8511 

8463 
8469 

8475 
8481 
8487 

8494  ! 
8500 
8506 

8512 

710 

8513 

8513 

8514 

8514  ;  8515 

85ib 

8516 

8522 
8528 
8535 
8541 
8547 
8553 
8559 
8565 
857i 

8517 

«5r7 

8518 

1 
2 
3 
4 
5 
6 
7 
* 
9 

8519 

8525 

853i 
8537 
8543 
8549 

%% 

8.,6i 

8567 

8519 

8525 
8532 
8538 
8544 
8550 
8556 
8562 
8568 

8520 
8526 
8532 
8538 
8544 
8550 

85f 
8562 
8569 

8521  i  8521 
8527  8527 

8533  :  8533 
8539   8539 
8545  :  8545 
8551  .  8552 

8557  !  8558 
8563  ;  8564 
8569  !  8570 

8522 
8528 
8534 
8540 
8546 
8552 
8558 
8564 
8570 

8523 
8529 
8535 
8541 
8547 
8553 
8559 
8565 
8572 

8524 
8530 
8536 
8542 
8548 
8554 
8560 
8566 
8572 

8524 
8530 
8536 
8542 
8549  i 
8555  ! 
8561  | 
8567 
8573 

720 

8573 

8574 

8575 

8575 

8576 

8576 

8577 

8578 

8578 

8579  ! 

1 
2 
3 
4 
5 
6 
7 
8 
9 

8579 
8585 
8591 

8597 
8603 
8609 
8615 
8621 
8627 

8580 
8586 
8592 

8598 
8604 
8610 
8616 
8622 
8628 

8581 
8587 
8593 

8599 
8605 
8611 
8617 
8623 
8628 

8581 
8587 
8593 

8599 
8605 

8611 
8617 
8623 
8629 

8582 
8588 
8594 
8600 
8606 
8612 
8618 
8624 
8630 

8582 
8588 
8594 
8600 
8606 
8612 
8618 
8624 
8630 

8583 
8589 
8595 
8601 
8607 
8613 
8619 
8625 
8631 

8584 
8590 
8596 
8602 
8608 
8614 
8620 
8625 
8631 

8584 
8590 
8596 
8602 
8608 
8614 
8620 
8626 
8632 

8585  i 
8591 
8597 
86o3 
8609 
8615 
8621 
8627 
8633 

730 

8633 

8634  |  8634 

8635    8636 

8636 

8637 

8637 

8638 

8639 

1 
2 
3 

'4 
5 
6 
7 
8 
9 

8639 
8645 
8651 

8657 
8663 
8669 

8675 
8681 
8686 

8640 
8646 
8652 
8658 
8663 
8669 

8675 
8681 
8687 

8640 
8646 
8652 
8658 
8664 
8670 
8676 
8682 
8688 

8641 
8647 
8653 

8659 
8665 
8671 
8676 
8682 
8688 

8642 
8647 
8653 
8659 
8665 
f  8671 

8677 
8683 
8689 

8642 
8648 
8654 

8660 
8666 
8672 
8678 
8684 
8689 

8643 
8649 

8655 
8661 
8666 
8672 
8678 
8684 
8690 

8643 
8649 

8655 
8661 
8667 
8673. 

8679 
8685 
8691 

8644 
8650 
8656 
8662 
8668 
8674 

8679 
8685 
8691 

8645 
8650 
8656 
8662 

8668 
8674 
8680 
8686 
8692 

740 

8692 

8693 

8693 

8694 

8695 

8695 

8696 

8696 

8697 

8698 

1 
2 
3 
4 

5 
6 

7 
8 
9 

8698 
8704 
8710 
8716 
8722 
8727 

8733 
8739 
8745 

8699 

8705 
8710 

8716 
8722 
8728 

8734 
8740 

8745 

8699 

8705 
8711 

8717 
8723 

8729 

8734 
8740 

8746 

8700 
8706 
8712 

8717 
8723 
8729 

8735 
8741 
8747 

8701 
8706 
8712 
8718 

8724 
8730 

8736 
8741 
8747 

8701 
8707 
8713 
8719 

8724 
8730 

8736 
8742 
8748 

8702 
8708 
8713 
8719 
8725 
8731 
8737 
8743 
8748 

8702 
8708 
8714 
8720 
8726 
8731 
8737 
8743 
8749 

8703 
8709 
8715 
8720 
8726 
8732 

8738 
8744 
8749 

8703 
8709 
8715 
8721 
8727 
8733 

8738 
8744 

8750 

750 

8751 

875i 

8752 

8752 

8753 

8754 

8754 

8755 

8755 

8756 

s   h  m  s                           s   h  m   s 

II     0    1     II                                                   II     Q      1     II 

700  =  011  40                       7000  =i  56  40 
710  =  o  ii  50                     7i°o  =i  58  20 

72O  =OI2    O                                72OO  —  2    O    O 

730  =  o  12  10                       73°°  =  2   i  4° 

740  —  0  12   20                                7400  =  2    3   20 
750  =  O  12   30                                7500  =  2    5    O 

TABLE  VI. 


19 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

0      1 

2 

3 

4 

5 

6      7 

8 

9 

I  750 

8751  I  8751 

8752    8752 

8753 

8754 

8754  i  8755 

8755 

8756 

1 
9 
3 

i    I 
5 
6 
7 

i 

9 

8756    8757 
8762  !  8763 
8768  |  8769 

8774    8774 

8779   !   8780 
8785   i   8786 
8791   i   8792 

8/97  i  8797 
8802  !  8803 

8758 
8763 

8769  ; 

8775  : 
8781 
8786  ; 
8792  i 
8798 
8804 

8758 
8764 
8770 

8775  ! 
8781 

8787 

8793 
8798 
8804 

8759 
8764 
8770 
8776 
8782 
8788 

8793 
8799 
8805 

8759 
8765 
877i 

8777 
8782 
8788 

8794 
8800 
8805 

8760    8760 
8766    8766 
8771    8772 
8777    8778 
8783    8783 
8789  i  8789 
8794  1  8795 
8800  j  8801 
8806    8806 

8761 
8767 
8773 
8778 

8784 
8790 

8796 
8801 
8807 

8762 
8767 
8773 
8779 
8785 
8790 
8796 
8802 
8808 

76O 

8808 

8809 

8814 

8820 
8826 
8832 

8837 
8843 

8849 
8854 
8860 

8809   8810 

8810 

8811 

8812 

8812 

8813 

8813 

1 
2 
3 
4 
5 
6 
7 
§ 
9 

8814 
8820 
8825 
8831 

8837 
8842 

8848 
8854 
8859 

8815   8816 

8821   8821 
8826   8827 
8832  :  8833 
8838  i  8838 
8843  |  8844 
8849  1  8850 
8855   8855 
8860   8861 

8816 
8822 
8828 

8833 
8839 
8845 
8850 
8856 
8862 

8817 
8822 
8828 

8834 
8839 
8845 
8851 
8856 
8862 

8817 
8823 
8829 
8834 
8840 
8846 
8851 
8857 
8863 

8818 
8824 
8829 

8835 
8841 
8846 
8852 
8858 
8863 

8818 
8824 
8830 

8835 
8841 

8847 
8852 
8858 
8864 

8819 

8825 
8830 
8836 
8842 
8847 

8853 
8859 
8864 

770 

8865    8865  • 

8866  -|  8867 

8867 

8868 

8868    8869 

8869 

8870 

1 
2 
3 
4 
5 
6 
7 
8 
9 

8871 
8876 
8882 
8887 
8893 
8899 

8904 
8910 
8915 

8871 
8877 
8882 

8888 
8894 
8899 
8905 
8910 
8916 

8872 
8877 
8883 
8889 
8894 
8900 
8905 
8911 
8916 

8872 
8878 
8883 
8889 
8895 
8900 

8906 
8911 
8917 

8873 
8878 
8884 
8890 

8895 
8901 

8906 
8912 
8918 

8873 
8879 
8885 
8890 
8896 
8901 
8907 
8913 
8918 

8874 
8880 
8885 
8891 
8896 
8902 
8908 

8913 
8919 

8874 
8880 
8886 
8891 
8897 
8903 
8908 
8914 
8919 

8875 
8881 
8886 
8892 
8898 
8903 
8909 
8914 
8920 

8876 
8881 
8887 
8892 
8898 
8904 
8909 

8915 
8920 

780 

8921 

8922 

8922 

8923 

8923 

8924 

8924  |  8925 

8925 

8926 

1 

2 
3 
4 

5 
6 

7 
8 
9 

8927 
8932 
8938 

8943 
8949 

8954 
8960 
8965 
8971 

8927 

& 

8944 
8949 

8955 
8960 
8966 
8971 

8928 
8933 
8939 
8944 
8950 
8955 
8961 
8966 
8972 

8928 
8934 
8939 

8945 
8950 
8956 
8961 
8967 
8972 

8929 

8934 
8940 

8945 
8951 
8956 

8962 
8967 
8973 

8929 

8935 
8940 

8946 
8951 
8957 
8963 
8968 
8974 

8930 

8935 
8941 

8946 
8952 
8958 
8963 
8969 
8974 

8930 
8936 
8942 

8947 
8953 
8958 
8964 
8969 
8975 

8931 
8937 
8942 

8948 
8953 
8959 
8964 
8970 

8975 

8932 
8937 
89.43 
8948 
8954 
8959 
8965 
8970 
8976 

790 

8976 

8977 

8977 

8978 

8978 

8979 

8980 

8980 

8981 

8981 

1 
2 
3 
4 

5 
6 

7 
8 
9 

8982 
8987 
8993 
8998 
9004 
9009 

9015 
9020 
9025 

8982 
8988 
8993 

8999 
9004 
9010 

9015 
9021 
9026 

8983 
8988 
8994 
8999 
9005 
9010 
9016 
9021 
9027 

8983 
8989 
8994 
9000 
9005 
9011 
9016 
9022 
9027 

8984 
8989 
8995 
9000 
9006 
9011 
9017 
9022 
9028 

8985 
8990 
8995 
9001 
9006 
9012 
9017 
9023 
9028 

8985 
8991 
8996 
9002 
90Q7 
9012 

9018 
9023 
9029 

8986 
8991 
8997 
9002 
9007 
9013 
9018 
9024 
9029 

8986 
8992 
8997 
9003 
9008 
9013 
9019 
9024 
9030 

8987 
8992 
8998 
9003 
9009 
9014 
9019 
9025 
9030 

SOO 

9031    9031 

9032 

9033 

9033 

9034 

9034 

9035 

9035 

9036 

s   h  m   s                           s   h  m   s 

II     Q    t    II                                                II     Q     1     It 

750  =  0  12   30                                7500  =  2    5    0 
760  —  O  12   40                                76OO  =  2    6   40 
770  =  0  12   50                                7700  =  2    8   20 
780  =  0  13    0                                7800  =  2   10    0 

790  =  0.13  10                       7900  =  2  ii  40 

800  =  0  13   20                                8000  =  2   13   20 

20 


TABLE  VI. 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

O 

123 

4 

5 

9034 

6 

9034 

? 

8 

9°35 

90*1 

9046 
9051 

9057 
9062 
9068 

9073  i 
9078 
9084 

9 

9036 

9041 
9047 
9052 

9057 
9063 
9068 
9074 
9079 
9084 

800 

9031 

9031 

9032 

9033 

9038  : 

9043  ; 
9049  i 

9054 
9060 
9065 
9070 
9076 
9081 

9033 

9035  i 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9036 
9042 
9047 

9°53 
9058 
9063 
9069 
9074 
9079 

9037  i 
9042  i 
9048 

9053 
9059 
9064 

9069 

9075 
9080  | 

9037 
9043  i 
9048 

9054 
9059 
9064 

9070 

9075 
9081 

9038 
9044 
9049 

9055 
9060 
9066 
9071 
9076 
9082 

9039 

9044 
9050 

9055 
9061 
9066 
9071 
9077 
9082 

9040  : 

9045 
9050  • 

9056 
9061 
9067 
9072 
9077 
9083 

9040 
9046 
9051 
9056 
9062 
9067 

9073 
9078 
9083 

810 

9085 

9085    9086 

9o86  |  9087 

9088 

9088, 

9089 

9089 

9090 

1 

2 
3 
4 
5 
6 
7 
8 
9 

9090 
9096 
9101 
9106 
9112 
9117 
9122 
9128 
9i33 

9091 
9096 
9101 
9107 
9112 
9117 
9123 
9128 
9133 

9091 
9097 
9102 
9107 

9"3 
9118 

9123 
9129 
9134 

9092 
9097 
9103 
9108 

9113 
9119 

9124 
9129 
9134 

9092 
9098 
9103 
9108 
9114 
9119 
9124 
9130 
9135 

9°93 
9098 
9104 
9109 
9114 
9120 
9125 
9130 
9135 

9093 
9099 
9104 
9109 

9"5 
9120 

9125 

9131 
9136 

9094 
9099 
9105 
9110 

9U5 
9121 

9126 
9131 
9137 

9094 
9100 
9105 
9111 
9116 
9121 
9126 
9132 
9i37 

9095 
9100 
9106 
9111 
9116 
9122 
9127 
9132 
9138 

820 

9138 

9139 

9139 

9140 

9140 

9141 

9141  |  9142 

9142 

9H3 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9H3 
9149 

9154 

9159 
9165 
9170 

9175 
9180 
9186 

9144 
9149 
9i55 
9160 
9165 
9170 
9176 
9181 
9186 

9144 
9i50 
9155 
9160 
9166 
9171 
9176 
9181 
9187 

9145 
9150 
9156 
9161 
9166 
9171 
9177 
9182 
9187 

9146 

9i5i 
9156 
9161 
9167 
9172 
9177 
9182 
9188 

9146 
9i5i 
9157 
9162 
9167 
9172 
9178 
9183 
9188 

9H7 
9152 
9*57 
9162 
9168 
91.73 
9178 
9183 
9189 

9H7 
9152 
9158 
9163 
9168 
9173 
9179 
9184 
9189 

9148 
9i53 
9158 
9163 
9169 
9J74 
9179 

9185 
9190 

9148 
9153 
9159 
9164 
9169 
9175 
9180 
9185 
9190 

830 

9191 

9191 

9192 

9192 

9193 

9193 

9194 

9194 

9195 

9195 

1 
2 
3 
*4 
5 
6 
7 
8 
9 

9196 
9201 
9206 
9212 

9217 
9222 

9227 

9232 
9238 

9197 
9202 
9207 
9212 
9217 
9223 
9228 

9233 
9238 

9197 
9202 
9207 
9213 
9218 
9223 
9228 
9233 
9239 

9198 
9203 
9208 
9213 
9218 
9224 
9229 
9234 
9239 

9198 
9203 
9209 

9214 
9219 
9224 
9229 

9235 
9240 

9199 
9204 
9209 
9214 
9219 
9225 
9230 

9235 
9240 

9199 
9204 
9210 

9215 
9220 
9225 
9230 
9236 
9241 

9200 
9205 
9210 

9215 
9221 
9226 
9231 
9236 
9241 

9200 
9205 
9211 
9216 
9221 
9226 
9231 

9237 
9242 

9201 
9206 
9211 
9216 
9222 
9227 
9232 

9237 
9242 

84O 

9243 

9243 

9244 

9244 

9245 

9245 

9246 

9246 

9247 

9247 

1 
2 
3 
4 
5 
6 

I 

8 
9 

9248 

9253 
9258 
9263 
9269 
9274 
9279 
9284 
9289 

9248 
9254 
9259 
9264 
9269 
9274 
9279 
9284 
9290 

9249 
9254 
9259 
9264 
9270 
9275 
9280 
9285 
9290 

9250 

9255 
9260 

9265 
9270 
9275 
9280 
9285 
9291 

9250 

9255 
9260 

9265 
9271 
9276 
9281 
9286 
9291 

9251 
9256 
9261 
9266 
9271 
9276 
9281 
9287 
9292 

9251 
9256 
9261 
9267 
9272 
9277 
9282 
9287 
9292 

9252 

9257 
9262 

9267 
9272 
9277 
9282 
9288 
9293 

9252 

9257 
9262 

9268 

9273 
9278 
9283 
9288 
9293 

9253 
9258 
9263 

9268 

92/3 
9278 
9283 
9289 
9294 

850 

9294 

9295 

9295 

9296 

9296 

9297 

9297 

9298 

9298 

9299 

s   /i  r/i   s                           s   h  m   s 

,,     0    1     II                                               H     0     I     'I 

800  =  0  13   20                                8000  =2   13   20 

810  =  o  13  30                       8100  =  2  15   o 
820  =  o  13  40                       8200  =  2  16  40 

830  =  0  13   50                                8300  =  2   18   20 
840  —  O  14    O                                8400  =  2   2O    O 
850  =  0  14   10                                8500  -2   21   40 

TABLE  VI. 


21 


Logarithms  of  Numbers  and  Small  Ares. 

No. 

0 

1 

2 

3 

4 

5 

6 

7     8 

9 

S5O 

9294 

9295 

9295    9296 

9296 

9297 

9297 

9298 

9298 

9299 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9299 

93°4 
93°9 

9315 
9320 

9325 
933° 
9335 
9340 

9300 

9305 
9310 

93»5 

9320 

9325 
933° 
9335 
9340 

9300 
93°5 
93" 
9316 
9321 
9326 

933i 
9336 
9341 

9301 
9306 
93" 
9316 

932i 
9326 

933i 
9336 
9341 

9301 
9306 
9312 

93'7 
9322 
9327 
9332 
9337 
9342 

9302 

93°7 
9312 

9317 
9322 

9327 
9332 
9337 
9342 

9302 
9307 
93!3 
93i8 
9323 
9328 

9333 
9338 
9343 

9303 
9308 

93i3 
93i8 
9323 
9328 

9333 
9338 
9343 

93°3 
9308 

93H 
9319 

9324 
9329 
9334 
9339 
9344 

93°4 
93°9 
93  H 
93*9 
9324 
9329 
9334 
9339 
9344 

860 

9345 

9345 

935i 
9356 
9361 

9366 
937i 
9376 

938i 
9386 

939i 

9346    9347 

9347 

9348 

9348 

9349 

9349 

9354 
9359 
9364 
9369 
9374 
9379 
9384 
9389 
9394 

9350 

1 
2 
3 
4 

5 
6 

7 
8 
9 

9350 
9355 
9360 

9365 
9370 
9375 
9380 
9385 
9390 

9351 
9356 
9361 

9366 
937i 
9376 

938i 
9386 

9391 

9352 
9357 
9362 

9367 
9372 
9377 
9382 
9387 
9392 

9352 
9357 
9362 

9367 
9372 
9377 
9382 
9387 
9392 

9353 
9358 
9363 
9368 

9373 
9378 

9383 
9388 

9393 

9353 
9358 
9363 
9368 
9373 
9378 

9383 
9388 
9393 

9354 
9359 
9364 
9369 
9374 
9379 
9384 
9389 
9394 

9355 
9360 

9365 
9370 

9375 
9380 

9385 
9390 
9395 

870 

9395 

9396 

9396 

9397 

9397 

9398 

9398 

9399 

9399 

9400 

1 
2 
3 
4 

5 

6 

7 
8 
9 

9400 

9405 
9410 

9415 
9420 

9425 
943° 
9435 
9440 

9401 
9406 
9411 
9416 
942i 
9426 

943° 
9435 
9440 

9401 
9406 
9411 
9416 
9421 
9426 

9431 
9436 
9441 

9402 
9407 
9412 

9417 
9422 
9427 

943  i 
9436 
9441 

9402 
9407 
9412 

9417 
9422 
9427 

9432 
9437 
9442 

9403 
9408 

9413 
9418 

9423 
9428 

9432 
9437 
9442 

9403 
9408 

9413 
9418 

9423 
9428 

9433 
9438 
9443 

9404 
9409 
9414 
9419 
9424 
9429 

9433 
943s 
9443 

9404 
9409 
9414 
9419 
9424 
9429 
9434 
9439 
9444 

9405 
9410 

9415 
9420 
9425 
943° 
9434 
9439 
9444 

i  880 

9445 

9445 

9446 

9446 

9447 

9447 

9448 

9448 

9449 

9449 

1 
2 
3 
4 
5 
6 
7 
8 
9 

945° 
9455 
9460 

9465 
9469 

9474 

9479 
9484 
9489 

9450 

9455 
9460 

9465 
9470 

9475 
9480 

9485 
9490 

9451 
9456 
9461 

9466 
9470 
9475 
9480 

9485 
9490 

9451 
9456 
9461 

9466 

9471 
9476 

948i 
9486 
9490 

9452 
9457 
9462 

9466 

9471 
9476 

9481 
9486 
9491 

9452 
9457 
9462 

9467 
9472 
9477 
9482 

9487 
9491 

9453 
9458 
9463 
9467 
9472 
9477 
9482 
9487 
9492 

9453 
9458 
9463 
9468 

9473 
9478 

9483 
9488 
9492 

9454 
9459 
9464 

9468 

9473 
9478 

9483 
948§ 

9493 

9454 
9459 
9464 

9469 
9474 
9479 
9484 
9489 
9493 

890 

9494 

9494 

9495 

9495 
9500 
9505 
95  10 

9515 
9520 

9525 
9529 
9534 
•  9539 

9496 

9496 

9497 

9497 

9498 

9498 

1 

2 

3 
4 

5 
6 

7 
8 
9 

9499 
95°4 
9509 
9513 
95i* 
9523 
9528 
9533 
9538 

9499 
95°4 
95°9 
95M 
95*9 
9524 
9528 
9533 
9538 

9500 
95°5 
95°9 
95*4 
95^9 
9524 
9529 
9534 
9539 

9501 
9506 

95  10 

9515 
9520 

9525 
9530 
9535 
9540 

95oi 
9506 

95" 
9516 
952i 
9526 

953° 
9535 
9540 

9502 
95°7 
95" 
95i6 
9521 
9526 

953i 
9536 
9540 

9502 

9507 
9512 

9517 
9522 
9526 

9531 
9536 
954i 

9503 
9508 
9512 

9517 
9522 
9527 
9532 
9537 
9541 

9503 
9508 

9513 
95i8 
9523 
9527 
9532 
9537 
9542 

900 

9542 

9543 

9543    9544 

9544 

9545 

9545 

9546 

9546 

9547 

j 

<  •  

ft   h  m   s                         s   h  m   s 

H     0    '     II                                                   II     0      1     II 

850  =  0  14   10                                8500  —  2.   21   40 
860  -  0  14   20                                8600  =  2   23   20 
870  =  0  14   30                                8700  =  2   25    0 
880  =  0  14  40                                8800  =  2   26   40 
890  =  0  14   50                                8900  -  2   28   20 

900  =  015   o                      9000  =  2  30   o 

22 


TABLE  VI. 


Logarithms  of  Numbers  and  Small  Arcs. 

No. 

0 

1 

2     3   |   4 

5     6     7   i   8 

9 

900 

9542 

9543 

9543    9544 

9544 

9545    9545 

9546  |  9546 

9547  | 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9547 
9552  i 
9557 
95j2  j 
9566 
9571 
9576 
958i 
9586 

9548 
9553 
9557 
9562 
9567 
9572 

9577 
958i 
9586 

9548 
9553 
9558 

9563 
9567 
9572 

9577 
9582 
9587 

9549 
9554 
9558 

9563 
9568 

9573 
9578 
9582 

9587 

9549 
9554 
9559 

9564 
9568 

9573 
9578 
9583 
9588 

955° 
9554 
9559 
9564 
9569 
9574 
9578 
9583 
9588 

9550 

9555 
9560 

9565 
9569 
9574 

9579 
9584 
9589 

955i 

9555  I 
9560 

9565  ' 
957° 
9575 
9579 
9584 
9589 

955i 
9556 
956i 
9566  1 
9570  1 
9575 
9580 
9585 
9589 

9594 

9552  i 
9556  ! 
956i 
9566 
957i 
9576 
958o  i 
9585 
9590 

91O 

9590 

9591 

959i 

9592 

9592 

9593 

9593 

9594 

9595  j 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9595 
9600 
9605 
9609 
9614 
9619 
9624 
9628 
9633 

9596 
9600 
9605 
9610 

9615 
9619 

9624 
9629 
9634 

9596 
9601 
9606 
9610 

9615 
9620 

9625 
9629 
9634 

9597 
9601 
9606 
9611 
9616 
9620 
9625 
9630 
•  9635 

9597 
9602 
9607 
9611 
9616 
9621 
9626 
9630 
9635 

9598 
9602 
9607 
9612 
9617 
9621 
9626 

963J 
9636 

9598 
9603 
9608 
9612 
9617 
9622 
9627 

9631 
9636 

9599 
9603 
9608 
9613 
9618 
9622 
9627 
9632 
9636 

9599 
9604 
9609 
9613 
9618 
9623 
9627 
9632 
9637 

9599 
9604 
9609 
9614 
9618 
9623 
9628 
9633 
9637 

920 

9638 

9638 

9639 

9639 

9640 

9640 

9641 

9641 

9642   9642  ! 

1 
2 
3 
4 

15 
6 
7 
§ 
9 

9643 
9647 
9652 

9657 
9661 
9666 
9671 

9675 
9680 

9643 
9648 
9652 

9657 
9662 
9667 
9671 
9676 
9681 

9644 
9648 
9653 

9658 
9662 
9667 

9672 
9676 
9681 

9644 
9649 
9653 
9658 
9663 
9668 
9672 
9677 
9682 

9644 
9649 
9654 

9659 
9663 
9668 

9673 
9677 
9682 

9645 
9650 

9654 

9659 
9664 
9668 

9673 
9678 
9682 

9645 
9650 

9655 
9660 
9664 
9669 
9674 
9678 
9683 

9646 
9651 
9655 
9660 
9665 
9669 
9674 
9679 
9683 

9646 
9651 
9656 
9660 
9665 
9670 

9675 
9679 
9684 

9647  i 
9652 
9656  ' 
9661  ! 

9666  ; 
9670  ! 

9675  ' 
9680  ; 

9684 

930 

9685 

9685 

9686 

9686 

9687 

9687 

9688 

9688 

9689    9689  ! 

1 
2 
3 
4 
5 
6 
7 
8 
9 

9690 
9694 
9699 

97°3 
9708 

97U 
9717 
9722 
9727 

9690 

9695 
9699 

9704 
9709 
9713 
9718 
9722 
9727 

9690 

9695 
9700 

9704 
9709 
97H 
9718 

9723 
9728 

9691 
9696 
9700 

9705 
9710 

97H 
9719 

9723 
9728 

9691 
9696 
9701 

9705 
9710 

9715 
9719 
9724 
9729 

9692 
9696 
9701 
9706 
9710 
9715 
9720 

9724 
9729 

9692 
9697 
9702 
9706 
9711 
9716 
9720 
9725 
9729 

9693 
9697 
9702 
9707 
9711 
9716 
9721 
9725 
973° 

9693 
9698 

9703 
9707 
9712 
9716 
9721 
9726 
9730 

9694  t 
9698  | 
9703  | 

9708  ; 
9712  i 
9717 
9722 
9726  j 

973  *  I 

940 

973i 

9732 

9732 

9733 

9733 

9734 

9734 

9735 

9735 

9735  ! 

1 
2 
3 

! 

6 

7 

§ 
9 

950 

9736 
9741 
9745 
9750 
9754 
9759 
9764 
9768 
9773 

9736 
974i 
9746 

975° 
9755 
9759 
9764 
9769 
9773 

9737 
974i 
9746 

975i 
9755 
9759 
9764 
9769 
9774 

9737 
9742 

9747 
975i 
9756 
9760 

9765 
9769 
9774 

9738 
9742 
9747 
9752 
9756 
9761 

9765 
9770 

9774 

9738 
9743 
9747 
9752 
9757 
9761 

9766 
9770 
9775 

9739 
9743 
9748 

9752 
9757 
9762 

9766 
9771 
9775 

9739 

9744 
9r48 

9753 
9758 
9762 

9767 
9771 
9776 

9740 
9744 
9749 

9753  , 
9758 
9763 
9767 
9772 
9776 

9740 

9745  i 
9749 
9754  ! 
9758  i 
9763  i 
9768 
9772 
9777 

9777 

9778 

9778 

9779 

9779 

9780 

9780 

9780 

9781 

978i 

s   h  m   s                        s   h  m   s 

II      O     1      II                                                            II      Q       1      II 

900  =  0150                        .        9000  =  2   30    0 

910  =  015  10                      9100  =  2  31  4o 

920  =  015   20                                9200  =  2   33   20 

930  =  015  30                       93°°  =  2  35   ° 

94O  =OI5   40                                94OO  =  2   36   40 
950  =  015   50                                9500  =  2   38   20 

-L^L. 

UAJJLJ   V  X.                                  ^U 

i  N^£IFQ'RT^ 

L.ogarit9ims  of  N  limbers  ami  Small  Arcs. 

fo. 

o 

9777 

"9782 
9786 
979  1 

9795 
9800 
9805 
9809 
9814 
9818 

1 

2 

3 

4 

5 

6     7 

§ 

9 

950 

9778 

9782 
9737 
9791 
9796 
9800 
9805 
9810 
9814 
9819 

9778 

9783 
9787 
9792 

9776 
9801 
9805 
9810 

9815 
9819 

9824 

9779 

9779 

978o 

9780 

9780 

9785 
9790 

9794 

9799 
9803 
9808 
9812 
9817 
9821 

9826 

978i 

9781 

1 
9 

3 
4 
& 

6 
7 

8 
9 

97*3 

9788 
9792 

9797 
9801 
9806 
9810 

98i5 
9820 

9784 
9788 

9793 

9797 
9802 
9806 
9811 

9815 
9820 

9825 

9784 
9789 
9793 
9798 
9802 
9807 
9811 
9816 
9820 

9785 
9789 
9794 
9798 
9803 
9807 
9812 
9816 
9821 

9785 
9790 

9795 

9813 
9817 
9822 

9786 
9790 
9795 
9800 
9804 
9809 

-9813 
9818 
9822 

960 

9823 

9827 
9832 
9836 
9841 

9845 
9850 

9854 

9859 

9803 

0823 

9824 

9825 

9825 

9826 

9827 

1 

2 
3 
4 
5 
6 
7 
§ 
9 

0828 
9832 
9837 
9841 
9846 
"9850 

9855 
98.S9 
9864 

9828 
9833 
9837 
9842 
9846 
9851 

9855 
9800 
9864 

9829 

9833 
9838 

9842 
9847 

9851 
9856 
9860 
9865 

9829 

9834 
9838 

9843 
9847 
9852 

9856 
9861 
9865 

9829 
9834 
9839 

9843 
9848 

9852 

9857 
9861 

9865 

9830 
9834 
9839 

9843 
9848 
9852  ' 

9857 
9861 
9866 

9830 
9835 
9839 
9844 
9848 
9853 

9857 
9862 
9866 

9831 
9f35 
9840 

9844 
9849 

9853 
9858 
9862 
9867 

9831 
9836 
9840 

9845 
9849 

9854 
9858 
9863 
9867 

97O 

9868 

9868 

9869    9869 

9870 

9870 

9870 

9871 

9871 

9872 

1 
2 
3 
4 
5 
6 
7 
i 
9 

9872 
9877 
9881 
9886 
9890 
9895 
9899 

9903 
9908 

9873 

$1 

9886 
9890 
9895 

9899 
9904 
9908 

9873 
9878 
9882 

9886 
9891 
9895 
9900 
9904 
9909 

9874 
9878 
9882 
9887 

9891 
9896 

9900 

9905 
9909 

9874 
9878 
9883 
9887 
9892 
9896 
9901 

9905 
9910 

9874 
9879 
9883 
9888 
9892 
9897 
9901 
9906 
9910 

9875 
9879 
9884 
9888 
9893 
9897 
9902 
9906 
9910 

9875 
9880 
9884 
9889 

9893 
9898 

9902 
9907 
9911 

9876 
9880 
9885 
9889 

9903 
9907 
9911 

9876 
9881 
9885 
9890 

9894 
9899 

9903 
9907 
9912 

9SO 

9912 

9913 

9913 

9914 

9914 

9914 

9915 

9915 

9916 

9916 

1 

2 
3 
4 

5 
6 

7 
8 

1    9 

9917 
9921 
9926 

993° 
9934 
9939 

9943 
9948 

9952 

9917 
9922 
9926 

993° 
9935 
9939 

9944 
9948 
9952 

9918 

9922 
9926 

9931 
9935 
9940 

9944 
9948 
9953 

9918 

9922 
9927 

993  i 
9936 
9940 

9944 
9949 
9953 

9918 

9923 
9927 

9932 
9936 
9941 

9945 
9949 
9954 

9919 

9923 
9928 

9932 
9937 
9941 

9945 
995° 
9954 

9919 
9924 
9928 

9933 
9937 
9941 

9946 
995° 
9955 

9920 
9924 
9929 

9933 
9937 
9942 
9946 
9951 
9955 

9920 
9925 
9929 

9933. 
9938 
9942 

9947 

995  i 
9955 

9921 
9925 
993° 
9934 
9938 
9943 
9947 
9952 
9956 

990 

9956 

9957 

9957 

9958 

9958 

9959 

9959 

9959 

9960 

9960 

1 

2 
3 
4 
5 
6 
7 
8 
9 

9961 

99^5 
9969 

9974 
9978 
9983 
9987 

9991 
9996 

9961 
9966 
9970 

9974 
9979 
9983 
9987 
9992 
9996 

9962 
9966 
9970 

9975 
9979 
9983 
9988 
9992 
9997 

9962 
9966 
9971 

9975 
9980 
9984 
9988 
9993 
9997 

9962 
9967 
9971 
9976 
9980  . 
9984 

9989 
9993 
9997 

9963 
9967 

9972 
9976 
9980 
9985 
9989 
9993 
9998 

9963 
9968 

9972  . 
9976 
998i 
9985 
9990 
9994 
9998 

9964 
9968 
9973 
9977 

9986 
9990 
9994 
9999 

9964 
9969 
9973 

9977 
9982 
9986 
9990 
9995 
9999 

9965 
9969 

9973 
9978 
9982 
9987 
9991 

9995 
oooo 

1000 

oooo 

oooo 

OOOI 

OOOI 

OOO2 

OOO2 

0003  1  0003 

0003 

0004 

s   h  m   $ 

il      Q    I     II 

^50  =  o  15  50 
960  =  o  16   o 
970  =  o  16  10 
986  =  o  16  20 
990  =  o  16  30 
1000  =  o  16  40 

*   /*  m   s 

II      0      1     II 

95OO  —  2   38   2O 
9600  =  2   40    O 
9700  =  2   41   40 

9800  —  2  43  20 
9900  —  2  45   o 

10000  —  2   46   40 

TABLE  \ 


Logarithms  of  Numbers:  1  to  1OO9. 

Ifo. 

0 

1 
I   2   3 

Proportional  Parts. 
4   ."»    «    7    N  •  9 

i 

1 

2  3 

4 

5  6 

7 

8 

9 

0 



oooo  3010  4771 

6021  6990  7782  8451  9031 

9542 

98 

196*294 

392490587 

685 

783 

88  1 

1 

OOOO 

0414:0792  1130 

1461  1761  2041  2305  2553 

2788 

30  60  90 

I2O  150  I8O2IO24O27O 

2 

3010 

3222  3424:3617 

3802  3979  4150  4314  44/2 

4624!  18;  35:  53 

71  88  io6|i241i42!iS9 

3 

4771 

4014  5052  5185 

5315  544i  55t>3  5682  5798  5911  13  25  38 

50  63  76 

88lOI  113 

4 

6021 

6128  623216335 

6435  6532  6628  6721  6812  6902  ;  10  19  29 

39!  48  58 

68 

78 

87 

flf 

6990 

7076^7160(7243 

7324  7404  7482  ,  7559  7634  7709   8  16 

24 

32  40  48 

56 

64 

72 

6 

7782 

7853  7924  7993 

8062  8129  8195  8261  8325 

8388 

7 

13 

20 

27  33  40 

47 

54 

60 

7 

8451 

§513  85/3:8633 

8692  8751  ;  8808  8865  i  8921 

8976 

6 

12 

18 

24 

29  35 

41 

47 

53 

§ 
9 

9031 
9542 

9085  9138  9191- 
959O  •  9638  9685 

9243  9294  9345 
9731  9777  9823 

9395  9445 
9868  ^912 

9494 
9956 

I 

IO 

9 

15 

20 

18 

25 
23 

28 

36 

$2 

37 

46  ! 
41 

10 

oooo 

0043  0086  0128 

0170 

0212 

0253 

0294 

°334 

0374 

4 

8 

12 

17  21 

25 

29 

33 

37 

0414 

045^  0492 

0531 

0569 

0607 

0645  0682  0719 

0755 

4 

8 

II 

15 

19 

21 

26 

10 

34 

2 

0792 

0828  0864  0899 

0934 

0969 

1004  1038  1072 

1106 

3 

7 

IO 

M 

17 

21 

24 

28 

31  1 

3 

"39 

1173  1206  1239 

1271 

M03 

1335  1367  1399 

143° 

3 

6 

10 

Ib 

19 

23 

26 

29 

4 

1461 

1492  1523 

1553 

1584  :  1614' 

1644  1673!  1703 

1732 

2 

6 

9 

12 

15 

18 

21 

24 

27 

5 

1761 

1790  1818  1847 

1875  1903 

1931  1959  ;  1987 

2014 

2 

6 

8 

II 

17 

20 

22 

25 

6 

2041 

2068  2095  2122 

2148  2175 

22OI  2227  2253 

2279 

2 

5 

8 

II 

13 

16 

18 

21 

24 

9 

2304 

-553 
2788 

233°  2355 
2577  2601 
2810  2833 

2380 
2625 
2856 

2405 
2648 
2878 

2430 
2672 
2900 

2455 
2095 

2923 

2480  ;  2504 
2718  2742 
2945  2967 

2529 
2765 
2989 

2 
2 
2 

5 
5 
4 

7 

7 
7 

10 

9 
9 

12 
12 
II 

15 

14 
13 

11 

16 

20 
19 

18 

22  ! 
21  ; 

20  : 

20 

3010 

3032 

3054 

3075 

3096 

3"8 

3*39 

3160  3181 

3201 

2 

4 

6 

8 

II 

13 

15 

17 

19 

1 

^222 

3243  3263 

3284 

3304 

3324 

3345 

336"5  3385 

3404 

2 

4 

6 

8 

IO 

12 

14 

16 

18 

2 

3424 

3444  3464 

3483 

1502 

3522 

354* 

356o  3579 

3598 

2 

4 

6 

8 

10 

12 

14 

M 

I7  i 

3 

36l/ 

3655 

3674 

3692 

37" 

3729 

3747  3766 

3/84 

2 

4 

6 

7 

9 

II 

13 

15 

17  j 

4 

^802 

^820 

3838 

3856 

3874 

3892 

3909  3927  3945 

3962 

2 

4 

5 

7 

9 

II 

12 

14 

16 

^ 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

2 

3 

5 

7 

9 

10 

12 

H 

15 

6 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

2 

3 

5 

7 

8 

10 

II 

13 

7 

43  H 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

2 

3 

5 

6 

8 

9 

II 

M 

H  ; 

8 

4472 

4487 

4502 

45i8 

14533 

4548 

45^4 

4579 

4594 

4609 

2 

3 

5 

b 

8 

9 

II 

12 

9 

4624 

4639 

4654 

4bb9 

4683 

4698 

4713 

4728 

4742 

4757 

I 

3 

4 

6 

7 

9 

IO 

12 

J3 

3O 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

I 

3 

4 

6 

7 

9 

10 

II 

13 

1 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

3 

4 

6 

7 

8 

10 

II 

12 

2 

5052 

5065 

5°79 

5092 

5  "9 

5132 

5H5 

5159 

5172 

3 

4 

5 

7 

8 

9 

II 

12 

3 

5185 

5198 

52" 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

3 

4 

5 

6  8 

9 

IO 

12 

4 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

3 

4 

5 

6  8 

9 

IO 

II 

5 

5453 

S478 

5490 

5502 

5515 

5527 

5539 

5551 

2 

4 

5 

6 

7 

9 

10 

II 

6 

5563 

5575 

5587 

5599 

5611 

5623 

5635 

5647 

5658 

5670 

2 

4 

5 

6 

7 

8 

10 

II 

1 

5682 
5798 

5694 
5809 

5705 
5821 

5832' 

5729 
5843 

5740 
5855 

5752 
5866 

5763  5775 
5877  5888 

5786 
5^99 

2 
2 

3 
3 

5 
5 

6 
6 

7 
7 

8 

8 

9 
9 

IO 
10 

9 

59" 

5922 

5933 

5944 

5955 

5977 

5988 

5999 

6010 

2 

3 

4 

5 

7 

8 

9 

10 

40 

6021 

6031 

6042 

6053 

6064 

6075 

6085 

6096 

6107 

6117 

2 

3 

4 

5 

6 

8 

9 

IO 

1 

6128 

6138 

6149 

6160 

6170 

6180 

6191 

6201 

6212 

6222 

2 

s 

4 

5 

6 

7 

8 

9 

3 

6232 
6335 

6243 
6345 

6253 
6355 

6263 
6365 

6274 
6375 

6284 
6385 

6294 
6395 

6304 
6405 

6314 
6415 

6325 
6425 

2 

2 

3 
3 

4 
4 

5 
5 

6 
6 

7 
7 

8  9 
8  9 

4 

6 

6435 
6532 
6628 

6444 
6542 
6637 

6454 
6646 

6464 
6561 
6656 

6474 

6571 
6665 

6484 
6580 
6675 

6493 
6590 
6684 

6503 
6693 

6609 
6702 

6522 
6618 
6712 

2 
2 

2 

3 
3 
3 

4 
4 
4 

5 
5 
5 

6 
6 
6 

7 
7 
7 

8 
8 
7 

9 

1 

6721 
6812 

6730 
6821 

6739 
6830 

St? 

6758 
6848 

6767 
68s  7 

6776 
6866 

6785 
6875 

6794 
6884 

6803 
6893 

2 

2 

3 

3 

4 
4 

5 
4 

5 
5 

6 
6 

7 
7 

8 
8 

9 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

2 

3 

4 

4 

5 

6 

7 

8 

50 

6990 

6998 

7007 

7016 

7024 

7033 

7042 

7050 

7059 

7067 

I 

2 

3 

3 

4 

5 

6 

7 

8 

TABLE  VII. 


25 


Logarithms  of  Numbers:  1  to  1OO9. 

No. 

0 

6990 

1 

2 

3 

4   5 

6 

* 

8 

9 

Proportional  Parts. 

1 

2 

3 

4 

5 

6 

5 

7 

6 

8 

9 

5O 

6998 

7007 

7016 

7024  7033  7042 

7050 

7059 

7067 

3 

3 

4 

7 
7 

I 

6 
6 
6 

6 
6 
6 

6 

6 
6 

5 

5 
5 
5 

5 
5 

8 

1 

3 
4 

6 
7 

9 

7076 
7160 

7243 

7324 
7404 
7482 

7559 

7634 
7709 

7084 
7168 
7251 

n-1  TJ 

7412 
7490 

7566 
7642 
7716 

7789 

7°93 

7177 

7259 

7340 
7419 
7497 

7574 
7649 

7723 

7101 
7185 
7267 

7348 

7427 

7505 

7582 
7657 
7731 

7110  7118 
7193  7202 
7275  |  7284 

7356  7364 
7435  :  7443 
7513;  7520 

7589  i  7597 
7664  7672 

773s  7745 

7126 
7210 
7*92 

7372 
745i 
7528 

7604 
7679 
7752 

7135 
7218 
7300 

7380 
7459 
7536 

7612 
7686 
7760 

7832 

7H3 

7226 

7308 

7388 
7466 

7543 
7619 
7694 
7767 

7152 
7235 
73l6 

7396 
7474 
755i 
7627 
7701 
7774 

i 
i 

i 
i 
i 

i 
i 

2 
2 

2 

2 
2 
2 

2 
I 
I 

3 

2 
2 

2 
2 
2 

2 

2 
2 

3 
3 
3 

3 

3 
3 

3 
3 
3 

4 

4 

4 

4 
4 
4 

4 
4 
4 

4 

4 
3 
3 

3 
3 
3 

3 
3 
3 

5 
5 
5 

5 
5 
5 

5 
4 
4 

6 
6 
6 

6 

5 

5 

5 
5 
5 

8 

7 
7 

7 
7 
7 

7 
7 

7 

6 

6 
6 
6 

6 
6 
6 

6 
6 
6 

60 

7782 

7796 

7803 

7810 

7818 

7825 

7839 

7846 

i 

i 

i 
i 

i 

i 
i 

i 
i 

I 

I 

I 
I 

I 

I 
I 

I 

I 
I 

2 

2 
2 

2 

2 
2 
2 

2 
2 
2 

3 

3 
3 
3 

3 
3 
3 

2 

4 

4 
4 
4 

4 
4 
4 

4 
4 
4 

5 

1 
2 
3 

4 

5 

6 

7 

8 
9 

7853 
7924 

7993 
8062 
8129 
8i95 
8261 

8325 
8388 

7860 

793i 
8000 

8069 
8136 
8202 

8267 
8331 
8395 

7868 

7938 
8007 

8075 
8142 
8209 

8274 
8338 
8401 

7875 
7945 
8014 

8082 
8149 
8215 

8280 

8344 
8407 

7882 
7952 
8021 

8089 
8156 
8222 

8287 

835i 
8414 

7889 

2£i 

8096 
8162 
8228 

8293 
8357 
8420 

7896 
7966 
8035 
8102 
8169 
8235 
8299 

8363 
8426 

7903 
7973 
8041 

8109 
8176 
8241 

8306 
8370 
8432 

7910 
7980 
8048 

8116 

8182 
8248 

8312 
8376 
8439 

7917 

7987 
8055 

8122 
8189 
8254 

8319 
8382 

8445 

5 
5 

5 

5 

5 
5 

5 
4 
4 

70 

8451 

8457 

8519 

8579 
8639 

8698 
8756 
8814 

8871 

8927 
8982 

8463 

8470 

8476 

8482 

8488 

8494 

8500 

8506 

i 

i 

i 
i 

i 

i 
i 
i 

I 

2 

2 

2 

2 

2 

2 

2 
2 

2 
2 
2 

3 

4 

4 

5 

5 
5 

5 
5 

4 
4 
4 

6 

5 
5 
'5 

5 
5 

5 

5 
5 
5 

1 

3 

4 

5 

6 

7 

9 

8513 
8573 
8633 
8692 

8751 
8808 

8865 
8921 
8976 

8525 
8585 
8645 

8704 
8762 
8820 

8876 
8932 
8987 

8531 

859i 
8651 

8710 
8768 
8825 

8882 
8938 
8993 

8537 
8597 
8657 
8716 

8774 
8831 

8887 

8943 
8998 

8543 
8603 
8663 

8722 
8779 
8837 

8893 
8949 
9004 

8549 
8609 
8669 

8727 
8785 
8842 

8899 

8954 
9009 

fl55 
8615 

8675 

8733 
8791 
8848 

8904 
8960 
9015 

8561 
8621 
8681 

8739 
8797 
8854 

8910 
8965 
9020 

8567 
8627 
8686 

8745 
8802 
8859 

8915 
8971 
9025 

I 
I 

I 

I 
I 

I 

I 
I 

I 

2 
2 
2 

2 
2 
2 

2 
2 
2 

3 
3 
3 

3 
3 
3 

3 
3 

4 
4 

4 

4 
3 
3 

3 
3 
3 

4 
4 
4 

4 
4 
4 

4 
4 
4 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

i 

I 

2 

2 

3 

3 

4 

4 

5 

1 

3 
4 

6 

8 
9 

90 

9085 
9138 
9191 

9243 
9294 

9345 

9395 
9445 
9494 

9090 

9H3 
9196 

9248 
9299 
935° 
9400 
945° 
9499 

9096 
9149 
9201 

9253 
93°4 
9355 

9405 
9455 
95°4 

9552 

9101 

9154 
9206 

9258 

93°9 
9360 

9410 
9460 
95°9 

9106 

9212 
9263 

9365 
9415 

9112 
9165 
9217 

9269 
9320 
9370 
9420 
9469 

9117 
9170 
9222 

9274 
9325 
9375 

9425 
9474 
9523 

9122 

9175 
9227 

9279 
933° 
9380 

943° 
9479 
9528 

9128 
9180 
9232 

9284 
9335 
9385 

9435 
9484 

9533 

9238 
9289 
9340 
9390 

9440 
9489 
9538 

9586 

i 

i 
i 

0 

o 

0 

o 

I 
I 

I 

I 

I 
I 

I 

2 
2 

2 

2 
2 
2 

I 
I 
I 

2 
2 

2 

2 
2 
2 

2 
2 
2 

3 
3 
3 

3 
3 
3 

2 
2 
2 

2 

2 
2 
2 

2 
2 
2 

2 

2 
2 

3 
3 
3 

3 
3 
3 

3 
3 
3 

4 
4 
4 

4 

4 
4 

3 
3 

3 

4 
4 
4 

4 
4 
4 

4 
4 
4 

5 

5 

5 
5 
5 

4 
4 
4 

4 

9542 

9547 

9557 

9562 

9566 

9571 

9576 

958i 

I 

2 

3 

3 

3 
3 

3 
3 

3 
3 

4 

3 

4 

6 

7 

9 

9590 
9638 
9685 

973  i 
9777 
9823 

9868 
9912 
9956 

9595 
9643 
9689 

9736 
9782 
9827 

9872 

9917 
9961 

9600 

9647 
9694 

9741 
9786 
9832 

9877 
9921 

9605 
9652 
9699 

9745 
9791 
9836 

9881 
9926 
9969 

9609 
9657 
9703 

975° 
9795 
9841 

9886 
993° 
9974 

9614 
9661 
9708 

9754 
9800 

9845 
9890 
9934 
9978 

9619 
9666 

9713 

9759 
9805 
9850 

9894 
9939 
9983 

9624 
9671 
9717 

9764 
9809 

9854 
9899 

9943 
9987 

9628 

9722 

9768 
9814 
9859 

9903 
9948 
9991 

9633 
9680 
9727 

9773 
9818 

9863 
9908 
9952 
9996 

0 

o 
o 

0 

o 
o 

o 

0 

o 

I 
I 

I 

I 
I 

I 

I 
I 
I 

I 
I 

I 

I 
I 

I 
I 

2 
2 
2 

2 
2 
2 

2 
2 
2 

3 
3 

3 

3 

3 
3 
3 

4 
4 

4 

4 
4 
4 

4 
4 
4 

4 
4 
4 

4 
4 
4 

4 
4 
4 

100 

oooo 

0004 

0009 

0013 

0017 

0022 

0026 

0030 

0035 

0039 

0 

I 

1 

2 

2 

3 

3 

4 

4 

26 


TABLE  VIII. 


Logarithmic  Sines,  Tangents,  and  Secants  to  every  Eighth  Point 

of  the  Compass. 

Degrees. 

Points. 

Sine. 

Cosecant. 

Tangent. 

Cotangent. 

Secant. 

Cosine. 

Points. 

Degrees. 

* 

0.0 

O 

—  oo 

+    00 

CO 

+    00 

10.0000 

10.0000 

8 

90.0 

1.4 

i 

8.3899 

11.6101 

8.3900 

1  1.  6100 

IO.OOOI 

9-9999 

I       88..  6 

91.9 

i 

8.6908 

11.3092 

8.6913 

11.3087 

10.0005 

9-9995 

I              87.2 

4.2 

I 

8.8667 

"-I333 

8.8678 

11.1322 

10.0012 

9.9988 

:              85.8 

5.6 

1 

8.9913 

11.0087 

8-9934 

11.0066 

10.0021 

9-9979 

1              84.4 

7.0 

I 

9.0878     10.9122 

9.0911 

10.9089 

10.0033 

9.9967 

1 

83.  0 

8.4 

I 

9.1665 

10.8335 

9.I7I3 

10.8287 

IO.OO47 

9-9953 

i 

81.6 

9.8 

t 

s 

9.2329     10.7671 

9.2394 

10.7606 

10.0064 

9.9936 

i 

80.2 

11.3 

1 

9.2902       10.7098 

9.2987 

10.7013 

10.0084 

9.9916 

7 

78.7 

12.7 

i 

9.3406    10.6594       9-3513 

10.6487 

IO.OIO7 

9-9893 

3 

77.3 

14.1 

i 

9.3856     10.6144        9-3988 

I  O.60  1  2 

10.0132 

9.9868 

f              75.9 

15.5 

i 

9.4260      10.5740 

9.4421 

10-5579 

10.0160 

9.9840 

f       74.5 

16.9 

i 

9.4628 

10.5372 

9.4819 

I0.5l8l 

10.0191 

9.9809 

i          !     73.1 

18.3 

i 

9.4965 

I0-5035 

9.5190 

10.4810 

10.0225 

9-9775 

||     71.7 

19.7 

I 

9-5275 

10.4725 

9-5537 

10.4463 

10.0262 

9.9738 

i              70.3 

21.1 

3. 

6 

9-5561 

10.4439 

9.5863 

10.4137 

10.0301 

9.9699 

i 

68.9 

22.5 

2 

9.5828 

10.4172 

9.6172 

10.3828 

10.0344 

9-9656 

6          67.5 

23.9 

i 

9.6077 

10.3923 

9.6467 

10-3533 

10.0390 

9.9610 

i       66.1 

25.3 

i 

9.6310 

10.3690 

9.6748 

10.3252 

10.0438 

9.9562 

f              64.7 

26.7 

I 

9.6528 

10.3472 

9.7019 

10.2981 

10.0490 

9.9510 

•        63  .  3 

28.1 

1 

9-6734 

10.3266 

9.7280 

10.2720 

10.0546 

9-9454 

i              61.9 

29.5 

1 

9.6928 

10.3072 

9.7532         10.2468 

10.0604 

9.9396 

1       60.5 

30.9 

1 

9.7111 

10.2889 

9-7777 

10.2223 

10.0666 

9-9334 

i 

4 

59.1 

32.3 

i 

9.7284 

10.2716 

9.8016 

10.1984 

10.0732 

9.9268 

i 

57.7 

33.7 

3 

9-7447 

IO-2553  i       9-8249 

10.1751 

10.0802 

9.9198 

5 

56.3 

35.2 

i 

9.7603 

10.2397 

9-8477 

10.1523 

10.0875 

9-9125 

t  |    54.8 

36.6 

i 

9-775° 

10.2250 

9.8702 

10.1298 

10.0952 

9.9048 

* 

53.4 

38.  0 

i 

9.7890 

IO.2IIO              9-8923 

10.1077 

10.1033 

9.8967 

i       52.0 

39.4     i 

9.8024 

10.1976           9-9142 

10.0858 

10.1118 

9.8882 

i 

50.6 

40.8 

f 

9.8150 

10.1850  j      9-9358 

10.0642 

10.1208 

9.8792 

I 

49.2 

42.2 

1 

9.8271       10.1729 

9-9573 

10.0427 

10.1302 

9.8698 

i 

47.8 

43.6 

1 

9.8386 

10.1614 

9.9787 

10.0213 

10.1401 

9.8599 

i 

46.  4^ 

45.0 

4 

9-8495 

10.1505         10.0000 

IO.OOOO 

10.1505 

9-8495 

4 

45.  0 

Degrees. 

Points. 

Cosine. 

Secant. 

Cotangent. 

Tangent. 

Cosecant. 

Sine. 

Points. 

Degrees. 

The  equivalents  in  Degrees  are  to  the  nearest  Tenth. 

TABLE  IX. 


27 


O0  =  O"  O™]                 l,og.  Sines,  Tangents,  and  Secants.       [llh  56m  =  179° 

' 

m    a 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     a 

/ 

O'.l 

1" 

O'.l 

1s 

o 
1 

AJI 

3 
4 
5 
6 
7 
8 
9 
1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

0    0 
4 

8 
12 
16 
020 
24 
28 
32 
36 
040 
44 
48 
52 
56 
1    0 
4 
8 
12 
16 
120 
24 
28 
32 
36 
140 
44 
48 
52 
56 
2    0 
4 
8 
12 
16 
220 
24 
28 
32 
36 
240 
44 
48 
52 
56 
3    0 
4 
8 
12 
16 
320 
24 
28 
32 
36 
34O 
44 
48 
52 
56 
4    0 

CO 

6.4637 
7648 
9408 
7-0658 

301 
176 
125 
97 

79 
67 

58 

5i 

40 

41.4 

37-8 
34-8 
32-2 
29.9 

28.0 
26.4 
24.8 

23-5 
22.3 

21.2 
2O.2 

19-3 
I8.5 
17.7 
17.0 
I6.4 
I5.8 
15.2 
14.7 

753 
440 
312 
242 
198 
167 

H5 
128 
114 
103 

94 
87 
80 

75 
70 
66 
62 
59 
56 

53 
50 
48 
46 
44 
42 
41 

3 

37 

+     CO 

13-5363 
2352 
0592 
12.9342 

—    CO 

6.4637 
7648 
9408 
7.0658 

301 
176 
125 

97 
79 
67 
58 

8 

41.4 
37-8 
34-8 
32.2 
29.9 
28.0 
26.4 
24.8 

23-5 
22.3 

21.2 

20.2 

19-3 

I8.5 
17.7 

17.0 
16.4 
I5.8 
15.2 
14.7 

753 
440 
312 
242 
198 
167 

;s 

114 
103 

94 

£ 

75 
70 
66 
62 
59 
56 

53 
5o 
48 
46 
44 
42 
41 
39 
38 
37 

+    CO 

13-5363 
2352 
0592 
12.9342 

10.0000 
oooo 
oooo 
oooo 
oooo 

10.0000 

oooo 
oooo 
oooo 
oooo 

60    0 
5956 
52 
48 
5944 
40 
36 
32 
28 
5924 
2O 
16 
12 
8 
59    4 
59    0 
5856 
52 
48 
5844 
40 
36 
32 
28 
5824 
20 
16 
12 
8 
58    4 
58    O 
5756 
52 
48 
5744 
40 
36 
32 
28 
5724 
20 
16 
12 
8 
57    4 
57    0 
56  56 
52 
48 
5644 
40 
36 
32 
28 
5624 
20 
16 
12 
8 
56    4 
56    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 

1 

6 
5 
4 

3 
2 

1 
O 

7.1627 
2419 
3088 
3668 
4180 

12.8373 
7581 

6912 

6332 
5820 

7.1627 
2419 
3088 
3668 
4180 

12.8373 
7581 

6912 

6332 

5820 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

74637 
5051 
5429 
5777 
6099 

12.5363 

4949 
457^ 
4223 
3901 

74637 
5051 
5429 
5777 
6099 

12.5363 

4949 
4571 
4223 
3901 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

7.6398 
6678 
6942 
7190 

7425 

12.3602 
3322 
3058 
2810 

2575 

7.6398 
6678 
6942 
7190 
7425 

12.3602 
3322 
3058 
2810 

2575 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

7.7648 

7859 
8061 

8255 
8439 

12.2352 
2141 
1939 
1745 
1561 

~i2~i'&3 
1213 
1049 
0891 
0739 

7.7648 
7860 
8062 
8255 
8439 

12.2352 
2140 
1938 
1745 
1561 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

7.8617 
8787 

8951 
9109 
9261 

7.8617 
8787 
8951 
9109 
9261 

12.1383 
1213 
1049 
0891 
0739 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

7.9408 

955i 
9689 
9822 
9952 
8.0078 

0200 

03'9 

0435 

0548 

!3 

25 

38 
50 
63 

Is 

IOI 

"3 

10 

19 

29 

39 

4-Q 

68 

78 
87 

8 
16 
24 
32 
40 
47 
55 
63 
7i 

i 

93 

25 
49 
73 

20 

$ 

17 

12.0592 
0449 
0311 
0178 
0048 

7.9409 

$& 

9823 
9952 

*3 

25 
38 

50 

63 

76 

88 

IOI 

"3 

10 

19 
29 
39 
49 

y 

78 
87 

8 
16 
24 
32 
40 
47 

3 

71 

31 
62 
93 

25 
49 
73 

20 

39 
58 

12.0591 
0449 
0311 
0177 
0048 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

10.0000 

oooo 
oooo 
oooo 
oooo 

11.9922 
9800 
9681 
9565 
9452 

8.0078 

O2OO 
0319 

°435 
0548 

8.06  58 
0765 
0870 
0972 
1072 

11.9922 
9800 
9681 
9565 
9452 

10.0000 

oooo 
oooo 
oooo 
oooo 

10.0000 

oooo 
oooo 
oooo 
oooo 

8.0658 
0765 
0870 
0972 

1072 

11.9342 

9235 
9130 
9028 
8928 

11.9342 

9235 
9130 
9028 
8928 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

8.1169 
1265 
1358 
145° 
1539 

11.8831 

8735 
8642 

8550 
8461 

8.II70 
1265 
1359 
145° 
1540 

11.8830 

8735 
8641 
8550 
8460 

IO.OOOO 

oooo 
oooo 
oooo 
oooo 

10.0000 

oooo 
oooo 
oooo 
oooo 

8.1627 
1713 

1797 
1880 

1961 

11.8373 
8287 
8203 
8120 
8039 

8.1627 

1713 

1798 
1880 
1962 

"-8373 

8287 
8202 
8120 
8038 

IO.OOOO 

oooo 
oooo 

OOOI 
OOOI 

IO.OOOO 

oooo 
oooo 
9-9999 
9999 

8.2041 
2119 
2196 

2271 

2346 

"•7959 

7881 
7804 
7729 
7654 

8.2041 
2I2O 
2196 

2272 
2346 

n-7959 

7880 
7804 
7728 
7654 

IO.OOOI 
OOOI 
OOOI 
OOOI 
OOOI 

9-9999 
9999 
9999 
9999 
9999 

8.2419 

11.7581 

8.2419 

11.7581 

IO.OOOI 

9.9999 

t  i 

m    s 

Cos. 

O'.l 

Sec. 

Cot. 

OM 

I8 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

9O°  -  6h  O™  ]                                                                                                          [  5^  56m  =  89° 

28 


TABLE  IX. 


!                                                                                                                                                                                                                                                                 '        i 

1°  -  oh  4ra  j                 Log.  Sines,  Tangents,  and  Secants.       [  llh  52m  =  178° 

' 

ni  s 

Sin. 

8.2419 
2490 
2561 
2630 
2699 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     9         ' 

O'.l 

1s 

O'.l 

I8 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
1O 
11 
12 
13 
!   14 
i   15 
16 
i   17 
!   18 
L  19 

2O 
21 

22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
!  38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
6O 

4    0 
4 

8 
12 
16 
420 
24 
28 
32 
36 
440 
44 
48 
52 
56 
5     0 
4 
8 
12 
16 
5  2O 
24 
28 
32 
36 
540 
44 
48 
52 
56 
6    0 
4 
8 
12 
16 
6  2O 
24 
28 
32 
36 
64O 
44 
4§ 
52 
56 
7    0 
4 
8 
12 
16 
720 
24 
28 
32 
36 
74O 
44 
48 
52 
56 
8    0 

7 

13 

20 

27 

34 
40 

47 
54 
60 

6 

12 
17 

23 

29 

35 

% 

52 

5 
10 

15 

20 
26 

$ 

41 
46 

5 
9 
H 

18 

23 
28 
32 
37 
4i 

J 

12 

16 
21 

25 
29 

33 
37 

i 

ii 
15 
19 
23 
27 

30 
34 

17 
33 

5° 

. 

15 
29 

43 

236 

38 

12 

23 

34 
10 

21 

31 

9 
19 
28 

~o7 

11.7581 
7510 
7439 
7370 
73°  i 
11.7234 
7108 
7102 
7038 
6975 

8.2419 
2491 
2562 
2631 
2700 

7 

'3 

20 

27 

34 
40 

47 
54 
60 

6 

12 

17 

23 
29 

35 

4' 
46 

52 

5 
10 

*5 

20 

26 

P 
$ 

5 
9 
H 

18 

3 

32 
37 
4i 

4 

8 

12 

16 

21 

25 
29 

33 
37 

4 
8 
ii 
15 
19 
23 
27 
3° 
34 

i/ 
33 
50 

15 
29 
43 

i 

38 

12 
23 

34 
10 

21 

3* 

9 
19 
28 

11.7581 

75  °9 
7438 
7369 
7300 

IO.OOOI 
OOOI 
OOOI 
OOOI 
OOOI 

9-9999 
9999 
9999 
9999 
9999 

56    O 
5556 
52 
48 
5544 
40 
36 
32 
28 
5524 
20 
16 
12 
8 
55    4 
55    O 
5456 
52 
48 
5444 
5440 
36 
32 
28 
5424 

O  A 

16 
12 

8 
54    4 
54    0 
5356 
52 
48 
5344 
40 
36 
32 
28 
5324 
20 
16 
12 
8 
53    4 
53    O 
5256 
52 
48 
5244 
40 
36 
32 
28 
5224 
20 
16 
12 
8 
52    4 
52    0 

6O 
59 
58 
57 
56 
55 
54 
53  ! 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40  I 
39 

38  ; 

37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21  i 
20 
19 
18 
17 
16 
15 
14  i 
13 
12 
11 

10 

9 

tt 

7 

5 
4 

3 
2 

1 

o  I 

8.2766 
2832 
2898 
2962 
3°25 

8.2767 

2833 
2899 
2963 
3026 

11.7233 

7167 
7101 

7037 
6974 

IO.OOOI 
OOOI 
OOOI 
OOOI 
OOOI 

9-9999 
9999 
9999 
9999 
9999 

8.3088 

315° 
3210 
3270 
3329 
8:3388' 
3445 
3502 
3558 
3613 

11.6912 
6850 
6790 
6730 
6671 

8.3089 

3J5° 
3211 

3271 
3330 

11.6911 
6850 
6789 

6729 
6670 

IO.OOOI 
OOOI 
OOOI 
OOOI 
OOOI 

9.9999 
9999 
9999 
9999 
9999 

11.6612 

6555 
6498 
6442 
6387 

8-3389 
3446 
3503 
3559 

11.6611 

6554 
6497 
6441 
6386 

IO.OOOI 
0001 
OOOI 
OOOI 
OOOI 

9-9999 
9999 
9999 
9999 
9999 

8.3668 
3722 

377=; 

3828 
3880 

11.6332 
6278 
6225 
6172 
6120 

8.3669 
3723 
3776 
3829 
3881 

11.6331 
6277 
6224 
6171 
6119 

IO.OOOI 
OOOI 
OOOI 
OOOI 
OOOI 

9-9999 
9999 
9999 
9999 
9999 

8-3931 
3982 
4032 
4082 
4131 

11.6069 
6018 
5968 
5918 

8-3932 
3983 
4033 
4083 

4132 

1  1.  6068 
6017 

5967 
5917 
5868 

IO.OOOI 
OOOI 
OOOI 
OOOI 
OOOI 

9-9999 
9999 
9999 
9999 
9999 

8.4179 

.4227 

4275 
432-2 
4368 

11.5821 

5773 

& 

5632 

8.4181 
4229 
4276 

4323 
4370 

11.5819 

5771 
5724 
5677 
563° 

IO.OOOI 
0002 
OOO2 
OOO2 
0002 

9-9999 
9998 
9998 
9998 
9998 

8.4414 
4459 
45°4 
4549 
4593 

n.5586 
554i 
5496 
545i 
5407 

8.4416 
4461 
4506 
455i 
4595 

ii-5584 
5539 
5494 
5449 
5405 

IO.OOO2 
OOO2 
0002 
OOO2 
OOO2 

9.9998 
9998 
9998 
9998 
9998 

8.4637 
4680 

4723 
4765 
4807 

8.4848 
4890 
493° 
4971 
5011 

"•5363 
5320 
5277 
5235 
5*93 

8.4638 
4682 

4725 
4767 
4809 

11.5362 

53i8 
5275 
5233 
5i9i 

IO.OOO2 
OOO2 
0002 
OOO2 
OOO2 

9.9998 
9998 
9998 
9998 
9998 

11.5152 
5110 
5070 
5029 
4989 

8.4851 
4892 
4933 
4973 
5013 

11.5149 
5108 
5067 
5027 
4987 

10.0002 
OOO2 
OOO2 
OO02 
0002 

9.9998 
9998 
9998 
9998 
9998 

8.5050 
5090 
5129 
5i67 
5206 

11.4950 
4910 
4871 
4833 
4794 

8.5053 
5092 

5*3' 

5170 
5208 

11.4947 
4908 
4869 
4830 
4792 

IO.OOO2 
OOO2 
0002 
OOO2 
OOO2 

9.9998 
9998 
9998 
9998 
9998 

8.5243 
5281 
53i8 
5355 
5392 

"•4757 
4719 
4682 

£§ 

8.5246 
5283 
532i 
5358 
5394 

"•4754 
4717 
4679 
4642 
4606 

10.0002 
OOO2 
OOO3 
0003 
0003 

9.9998 
9998 
9997 
9997 
9997 

8.5428 

11.4572 

8.5431 

11.4569 

IO.OOO3 

9.9997 

/ 

m  a 

COB. 

O'.l 

Sec. 

Cot. 

O'.l 

!• 

Tan. 

Cosec. 

Sin. 

m    a 

/ 

Diff. 

Diff. 

91°  =  6U  4m  ]                                                                                                         [  5h  52m  =  88° 

TABLE  IX. 


;  2^  =  O"  8™  ]                 Log:.  Sines,  Tangents,  and  decants.       [  ll1  48'»  =  177° 

O 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
2O 
21 
22 
23 
24 
25 
26 
27 
28 
29 
3O 
i  31 
i  32 
33 
|  34 
!  35 
36 
37 
!  38 
!  39 
i  40 
1  41 
!  42 
1  43 
|  44 
i  45 
i46 
|47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
6O 

m     s 

8     0 
4 
8 
12 
16 
8  2O 
24 
28 
32 
36 
84O 
44 
48 
52 
56 
9    0 
4 
8 
12 
16 
920 
24 
28 
32 
36 
94O 
44 
48 
52 
56 
1O    O 
4 
8 
12 
16 
1020 
24 
28 
32 
36 
1O4O 
44 
48 
52 
56 
11    0 
4 
8 
12 
16 
11  2O 
24 
28 
32 
36 
1140 
44 
48 
52 
56 
12    0 

Sin. 

8.5428 
5464 
5500 
5535 
557i 

Diff. 

Cosec. 

Tan. 

Di 
07.1 

ff. 
1s 

Cot. 

Sec. 

10.0003 
0003 
'  0003 
0003 
0003 

Cos. 

m    s 

' 

O'.l 

1s 

4 
7 
n 

H 
17 

21 

% 

31 

3 
6 

10 

J3 
16 
19 

22 
26 
29 

6 
9 

12 

21 

24 

27 

6 
8 
ii 

H 
17 

20 

22 
25 

3 
10 

J3 

16 

18 

21 

23 

3 

5 

7 

10 
12 
15 

17 
20 
22 

9 
if 

27 

8 
16 
24 

8 
15 

22 

7 
H 

21 

7 
13 
19 

6 

12 
19 

II-4572 

4536 
4500 

4465 
4429 

8-5431 
5467 
55°3 
5538 
5573 

\ 

ii 
H 

17 

21 

24 

28 

31 

6 

10 

*3 
16 
19 

22 
26 
29 

I 

9 

12 
15 

18 

21 

24 
27 

6 
8 
n 

H 
17 

20 

22 
25 

3 
IO 

13 

16 
18 

21 

23 

3 

5 
7 

IO 
12 
15 
17 

,    2O 
22 

i 

27 

8 
16 
24 

8 
15 

22 

7 
14 

21 

7 
13 
19 

6 

12 

19 

11.4569 
4533 
4497 
4462 
4427 

9-9997 
9997 
9997 
9997 
9997 

52    0 
51  56 
52 
48 
5144 
40 
36 
32 
28 
5124 
2O 
16 
12 
8 
51    4 
51    O 
5056 
52 
48 
5044 
40 
36 
32 
28 
5O24 
2O 
16 
12 
8 
50    4 
50    0 
4956 
52 
48 
4944 
40 
36 
32 
28 
4924 
2O 
16 
12 
8 
49    4 
49    0 
4856 
52 
48 
4844 
40 
36 
32 
28 
48  24 
2O 
16 
12 
8 
48    4 
48    0 

6O 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 

3 
2 

1 
0 

8.5605 
5640 
5674 
57o8 
5742 

n-4395 
4360 
4326 
4292 
4258 

8.5608 
5643 
5677 
57" 
5745 

11.4392 
4357 
4323 
4289 

4255 

10.0003 
0003 
0003 
0003 
0003 
10.0003 
0003 
0003 
0003 
0003 

9-9997 
9997 
9997 
9997 
9997 

8.5776 
5809 
5842 
5875 
59°7 

11.4224 
4191 
4158 
4125 
4093 

8-5779 

5812 

l*& 

5878 

59" 

11.4221 

4188 

4155 

4122 
4089 

9-9997 
9997 
9997 
9997 
9997 

8-5939 
5972 
6003 

6035 
6066 

11.4061 
4028 
3997 
3965 
3934 

8-5943 
5975 
6007 
6038 
6070 

11.4057 
4025 

3993 
3962 

393° 

10.0003 
0003 
0003 
0003 
0004 

9-9997 
9997 
9997 
9997 
9996 
9.9996 
9996 
9996 
9996 
9996 

8.6097 
6128 

6i59 
6189 
6220 

11.3903 

3872 

3841 
3811 
378o 

8.6101 
6132 
6163 
6193 
6223 
8.6254 
6283 
6313 
6343 
6372 

11.3899 

3868 

3837 
3807 

3777 

10.0004 
0004 
0004 
0004 
0004 

8.6250 
6279 
6309 

6339 
6368 

"•3750 
372i 
3691 
3661 
3632 

11.3746 

3717 
3687 

3657 
3628 

10.0004 
0004 
0004 
0004 
0004 

9-9996 
9996 
9996 
9996 
9996 

8.6397 
6426 

6454 
6483 
6511 

11.3603 
3574 
3546 
3517 
3489 

8.6401 
6430 

6459 
6487 

6515 

11  -3599 
3570 
354i 
3513 

3485 

10.0004 
0004 
0004 
0004 
0004 

9-9996 
9996 
9996 
9996 
9996 

8-6539 
6567 

6595 
6622 
6650 

11.3461 
3433 
3405 
3378 
335° 

8.6544 
6571 

6§97 
6654 

1  1  -3456 
3429 
340i 
3373 
3346 

10.0004 
0004 
0005 
0005 
0005 

9.9996 
9996 
9995 
9995 
9995 

8.6677 
6704 
6731 
6758 
6784 

11-3323 

3296 
3269 
3242 
3216 

8.6682 
6709 
6736 
6762 
6789 

11-3318 
3291 
3264 

3238 
3211 

10.0005 
0005 
0005 
0005 
0005 

9-9995 
9995 
9995 
9995 
_9991 
9-9995 
9995 
9995 
9995 
9995 

8.6810 
6837 
6863 
6889 
6914 

11.3190 
3163 
3137 
3111 

3086 

8.6815 
6842 
6868 
6894 
6920 

11.3185 
3158 
3132 
3106 
3080 

10.0005 
0005 
0005 
0005 
0005 

8.6940 
6965 
6991 
7016 
.7041 

11.3060 

3°35 
3009 
2984 
2959 

8-6945 
697J 
6996 
7021 
7046 

"•3055 
3029 
3004 
2979 
2954 
11.2929 
2904 
2879 

2855 
2830 

10.0005 
0005 
0005 
0005 
0006 

9-9995 
9995 
9995 
9995 
9994 

8.7066 
7090 

7H5 

7140 
7164 

11.2934 
2910 
2885 
2860 
2836 

8.7071 
7096 
7121 

7H5 
7170 

10.0006 
0006 
0006 
0006 
0006 

9-9994 
9994 
9994 
9994 
9994 

8.7188 

11.2812 

8.7194 

11.2806 

10.0006 

9-9994 

/ 

1.    L 

m     s 

! 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

I8 

Tan. 

Cosec. 

Sin. 

111      S 

/ 

Diflf. 

Diff. 

|  92a  =  6h  8m  ]                                                                                                          [  5h  4gm  _  8.y° 

30 


TABLE  IX. 


33  =  Oh  12m  ]              L.O&.  Sines,  Tangents,  and  Secants.       [  llh  44ni  =  176° 

• 

111    a 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.       Cos. 

m    s 

O'.l  !  1s 

O'.l     1s 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 

45 
40 
147 
49 
49 
50 
51 
52 
53 
54 
!  55 
56 
57 
58 
59 
60 

12    0 
4 

8 
12 
16 
12  2O 
24 
28 
32 
36 
12  4O 
44 
48 
52 
56 
13    O 
4 
8 
12 
16 
13  2O 
24 
28 
32 
36 
13  4O 
44 
48 
52 
56 
14    O 
4 
8 
12 
16 
14  2O 
24 
28 
32 
36 
1440 
44 
48 
52 
56 
15    0 
4 
8 
12 
16 
1520* 
24 
28 
32 
36 
15  4O 
44 
48 
52 
56 
16    0 

8.7188 
7212 
7236 
7260 
7283 

2 

5 

7 
9 

12 

16 
18 

21 

2 

4 
7 
9 
ii 

II 

18 

20 

2 

4 
6 
8 
ii 

15 
17 
19 

2 

1 

8 

10 
12 

\l 

18 

2 

1 

8 

10 
12 
14 
15 
17 

2 

6 

7 

9 
ii 

13 
15 
17 

6 

'  12 
17 

6 
ii 
16 

5 
ii 
16 

5 

10 

15 

5 
10 

5 
9 

11.2812 
2788 
2764 
2740 
2717 

8.7194 
7218 
7242 
7266 
7290 

2 

5 
7 
9 

12 

16 

18 

21 

2 

4 

7 
9 
u 

i 

20 
2 

1 

8 
ii 
13 
15 
17 
19 

2 

4 
6 
8 

10 
12 

14 

16 
18 

2 

4 
6 
8 

10 
12 

15 
17 

2 

4 
6 

7 

9 
ii 

13 
15 
17 

6 

12 
17 

6 
ii 
,6 

5 
ii 

16 

5 

10 

15 

5 
10 

14 

5 
9 

14 

11.2806 
2782 
2758 
2734 
2710 

10.0006    9.9994 
0006       9994 
0006       9994 
0006       9994 
0006       9994 

48    O 
4756 
52 
48 
4744 
4O 
36 
32 
28 
47  24 
2O 
16 
12 
8 
47    4 
47    0 
4656 
52 
48 
4644 
40 
36 
32 
28 
4624 
20 
16 
12 
8 
46    4 
46    0 
4556 
52 
48 
4544 
40 
36 
32 
28 
4524 
2O 
16 
12 
8 
45    4 
45    0 
4456 
52 
48 
4444 
40 
36 
32 
28 
4424 
20 
16 
12 
8 
44    4 
44    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51  . 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18  i 
17 

15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4  1 

2 

1 
0 

8.7307 
733° 
7354 
7377 
7400 

11.2693 
2670 
2646 
2623 
2600 

87313 
7337 
7360 

7383 
7406 

11.2687 
2663 
2640 
2617 
2594 

10.0006    9.9994 
0006       9994 
0006       9994 
0006       9994 
0007       9993 

8.7423 

7445 
7468 

749i 
7513 

11.2577 

2555 
2532 
2509 

2487 

8-7429 
7452 
7475 
7497 
7520 

11.2571 
2548 
2525 
2503 
2480 

10.0007 
0007 
0007 
0007 
0007 

9-9993 
9993 
9993 
9993 
9993 

8-7535 
7557 

7602 
7623 

11.2465 

2443 
2420 

2398 
23/7 

8.7542 
7565 
7587 
7609 

7631 

11.2458 
2435 
2413 
2391 
2369 

10.0007 
0007 
0007 
0007 
0007 

9-9993 
9993 
9993 
9993 
9993 

8.7645 
7667 
7688 
7710 
773  l 
8.7752 
7773 
7794 

7836 

"•2355 
2333 
2312 
2290 
2269 

8.7652 
7674 
7696 
7717 
7739 

11.2348 
2326 
2304 
2283 
2261 

10.0007 
0007 
0007 
0008 
0008 

9-9993 
9993 
9993 
9992 
9992 

11.2248 
2227 
2206 
2185 
2164 

8.7760 

778i 
7802 
7823 
7844 

11.2240 
2219 
2198 
2177 
2156 

10.0008 
0008 
0008 
0008 
0008 

9.9992 
9992 
9992 
9992 
9992 

8.7857 
7877 

7898 
7918 

7939 

11.2143 
2123 

2IO2 
2082 
2061 

8.7865 
7886 
7906 
7927 
7947 

".2135 
2114 
2094 
2073 
2053 

10.0008 
0008 
0008 
0008 
0008 

9.9992 
9992 
9992 
9992 
9992 

8-7959 
7979 
7999 
8019 
8039 

II.2O4I 
2021 
2001 
I98l 
I96l 

8.7967 
7988 
8008 
8028 
8048 

11.2033 

2OI2 
1992 
1972 
I952 

10.0009 
0009 
0009 
0009 
0009 

9.9991 
9991 
9991 
9991 
9991 

8.8059 
8078 
•8098 
8117 
8i37 

II.I94I 

1922 

1863 

8.8067 
8087 
8107 
8126 
8146 

H.I933 
1913 
1893 
1874 

185.4 

10.0009 
0009 
0009 
0009 
0009 

9.9991 
9991 
9991 
9991 
9991 

8.8156 

8175 
8194 
8213 
8232 

11.1844 
1825 
1806 

1787 
1768 

8.8165 
8185 
8204 
8223 
8242 

II.I835 

1796 
1777 

1758 

10.0009 
0009 
0009 

OOIO 
OOIO 

9.9991 
9991 
9991 
9990 
9990 

8.8251 
8270 
8289 
8307 
8326 

11.1749 

173° 
I7II 
1693 
1674 

8.8261 
8280 
8299 
8317 
8336 

".1739 
1720 
1701 
1683 
1664 

IO.OOIO 
OOIO 
OOIO 
OOIO 
OOIO 

9.9990 
9990 
9990 
9990 
9990 

8.8345 

8363 
8381 
8400 
8418 

II.I655 
1637 
1619 
1600 
1582 

8.8355 
8373 
8392 
8410 
8428 

II.I645 
1627 
1608 
1590 
1572 

IO.OOIO 
OOIO 
OOIO 
OOIO 
OOIO 

9.9990 
9990 
9990 
9990 
9990 

8.8436 

11.1564 

8.8446 

11.1554 

IO.OOII 

9.9989 

' 

m      a 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l  |  1« 

Tan. 

Cosec. 

Sin. 

m     s 

'  j 

Diff. 

Diff. 

93°  =  6h  12m  ]                                                                                                      [  5h  44m  =  86°  | 

TABLE  IX. 


31 


40r=oh  1611]               Log:.  Sines,  Tangents,  and  Secants.       [llh  4OU1  =  175° 

i 

in    8 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     s 

/ 

0.1     1 

O'.l 

1s 

O 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
|  14 
!  15 
16 
17 
i  18 
19 
20 
21 
22 
i  23 
|  24 

!  25 
26 
!  27 
28 
29 
30 
31 
32 
33 
34 
35 
!  36 
i  37 
i  38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
151 
|52 
53 
54 
55 
56 
57 
58 
59 
60 

16     O 
1 

8 
12 

16 

'     1620 

24 
28 
32 
36 

164O 
44 

48 
52 
56 
17    0 
4 
8 
12 
16 
1720 
24 
28 
32 
36 
17  4O 
44 
48 
52 
56 
18    0 
4 
8 
12 
16 
1820 
24 
28 
32 
36 
18  4O 
44 
48 
52 
56 
19    0 
4 
8 
12 
16 
1920 
24 
28 
32 
36 
1940 
44 
48 
52 
56 
20    0 

8.8436 

8454 
8472 
8490 

8508 

"8.8525 

8543 
8560 

8578 
8595 
8.8613 
8630 
8647 
8665 
8682 
8.8699 
8716 

8733 
8749 
8766 
""8^8783" 

8799 
8816 

8833 
8849 

2       4 
4       9 
5     13 

7 

9 
ii 

12 
H 

1  6 

2         4 

3       9 
5      '3 

7 

9 
10 

12 

H 
15 

2         4 

.   3       8 

5       :        12 

7  . 
8 

TO 

ii' 
J3 

15 

2       4 
3       8 

1     " 
8 

9 
ii 

13 
14 

2   i      4 

3       8 

Ii" 

8 

9 
ii 

12 

H 

i       4 
3       7 
4     " 
6 

7 
9  i 

10 
12 
13 

11.1564 
1546 
1528 
1510 
J492 
"•W5 
1457 
1440 
1422 
1405 
1  1~  138  7 
1370 
1353 
J335 

_iM 

11.1301 

1284 
1267 
1251 

J234 
11.1217 

I2OI 
1184 
1167 

"5i 

8.8446 
8465 

8483 
8501 
8518 

2 

4 

5 

7 

9 
ii 

12 

14 

16 

2 

3 
5 
7 
9 

10 
12 
H 
15 

2 

3 

7 
8 

10 

ii 
13 
15 

2 

3 
8 

Q 

•    II 
13 
14 

2 

3 

6 
8 

9 
ii 

12 

H 

i 
3 
4 
6 

7 
9 

10 
12 
13 

4 
9 
13 

1 
I 

4 
9 
13 

i 

12 

4 
8 

12 

8 
ii 

4 
7 
ii 

11.1554 
1535 
1517 

1499 
1482 

IO.OOII 
001  1 
OOII 
OOII 
OOII 

9.9989 
9989 
9989 
9989 
9989 

44    0 
4356 
52 
48 
4344 
40 
36 
32 
28 
4324 
2O 
16 
12 
8 
43    4 
43    O 
4256 
52 
48 
4244 
40 
36 
32 
28 
4224 
20 
16 
12 
8 
42    4 
42    O 
4156 
52 
48 
4144 
40 
36 
32 
28 
4124 
20 
16 
12 
8 
41    4 
41    O 
4056 
52 
48 
4044 
40 
36 
32 
28 
4O24 
20 
16 
12 
8 
40    4 
40    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49  S 
48 
47 
46 
45 
44 
43 
42  ! 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

8.8536 
8554 
8572 
8589 
8607 
8.8624 
8642 
8659 
8676 
8694 
8.8711 
8728 

8745 
8762 

8778 
8:8795 
8812 
8829 
8845 
8862 

11.1464 
1446 
1428 
1411 
1393 
11.1376 
1358 
H4i 
i324 
J3o6 

11.1289 
1272 

1255 
1238 

1222 

IO.OOII 
OOII 
OOII 
OOII 
OOII 
IO.OOII 
0012 
OOI2 
OOI2 
OOI2 
IO.OOI2 
OOI2 
0012 
OOI2 
0012 

9.9989 
9989 
9989 
9989 
__99?9_ 
9.9989 
9988 
9988 
9988 
9988 
9.9988 
9988 
9988 
9988 
__?9_8S 
9.9988 

9987 
9987 
9987 
9987 

II.I2O5 

1188 
1171 

H55 
1138 

IO.OOI2 
OOI3 
OOI3 
0013 
OOI3 

8.8865 
8882 
8898 
8914 
8930 
8.8946 
8962 
8978 
8994 
9010 

Il-ll35 
1118 

IIO2 
1086 
1070 

8.8878 

8895 
8911 
8927 
8944 

11.1122 
1105 
I089 
1073 
1056 

10.0013 
OOI3 
OOI3 
OOI3 
OOI3 

9.9987 
9987 
9987 
9987 
9987 

11.1054 
1038 
IO22 
IOO6 
0990 

8.8960 
8976 
8992 
9008 
9024 

11.1040 
IO24 
1008 
0992 
0976 

IO.OOI3 
0014 
OOI4 
0014 
OOI4 

9.9987 
9986 
9986 
9986 
9986 
9.9986 
9986 
9986 
9986 
9986 

8.9026 
9042 
9057 
9073 
9089 

11.0974 
0958 

°943 
0927 
0911 

8.9040 
9056 
9071 
9087 
9103 

11.0960 
0944 
0929 
0913 
0897 

IO.OOI4 
OOI4 
OOI4 
OOI4 
0014 

8.9104 
9119 
9135 
9i50 
9165 

11.0896 
0881 
0865 
0850 
0835 

8.9118 

9r34 
9150 
9165 
9180 

11.0882 
0866 
0850 

0835 
O82O 

IO.OOI4 
0015 
OOI5 
0015 
OOI5 

9.9986 
9985 
9985 
9985 
9985 

8.9181 
9196 
9211 
9226 
9241 
8.9256 
9271 
9286 
9301 
9315 

11.0819 
0804 
0789 
0774 
°759 

8.9196 
9211 
9226 
9241 
9256 

11.0804 
0789 
0774 

0759 
0744 

IO.OOI5 
0015 
OOI5 
0015 
0015 

9.9985 
9985 
9985 
9985 
9985 

11.0744 
0729 
0714 
0699 
0685 

8.9272 
9287 
9302 
9316 

933  i 

11.0728 
0713 
0698 
0684 
0669 

11.0654 
0639 
.  0624 
o6lO 
0595 

10.0015 
OOl6 
OOl6 
OOl6 
OOl6 

9.9985 

9984 
9984 

9984 
9984 

8.9330 
9345 
9359 
9374 
_9_388 

8.9403 

11.0670 

0655 
0641 
0626 
0612 

8.9346 
9361 
9376 
9390 
9405 

IO.OOI6 
OOl6 
00l6 
OOl6 
OOl6 

9.9984 
9984 

9984 
9984 

9984 

11.0597 

8.9420 

11.0580 

IO.OOI7 

9.9983 

/ 

m    s 

Cos. 

O'.l     1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

949  =  6h  16m  ]                                                                                                       [  5U  4Om  =  85° 

32 


TABLE  IX. 


i  5°  =  On  2Om  ]              L,og.  Sines,  Tangents,  and  Secants.       [  llh  36m  =  1743 

/ 

0 
1 
2 
3 
4 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
i54 
55 
56 
57 
58 
59 
60 

in     s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.      Cos. 

m     s 

60 
59 
58  ; 
57 
56  | 
55 
54  i 
53 
52 
51 

5O 
49 

48  ; 

47 
46 

45  ; 

41 
43 
42 
4B 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28  i 
27  ; 
26 
25 
24 
23  i 
22 
21 
2O 
19 
18 
17 
16 
15 
14  i 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

0.1 

i 
3 
4 
6 
7 
8 

10 

ii 
13 

i 

3 
4 
6 

I 

10 

ii 
13 

i 

3 
4 
5 

I 

9 
10 

12 

I 

3 
4 

5 

9 
10 

12 

I 

3 
4 
5 
6 
8 
9 

% 

i 

2 

4 
5 
6 

8 
10 
ii 

1" 

4 
7 
ii 

4 
7 
ii 

I 

10 

\ 

10 
10 

9 

O'.l 

i 
3 

A 

6 

•j 

8 

10 

ii 
I3 

i 
3 
4 
6 

8 
10 
ii 
13 

i 

3 
4 
5 

Q 

10 

12 

I 

3 
4 
5 

8 
9 

10 
12 

I 

3 
4 
5 
6 
8 
9 

10 
12 

I 

2 

4 
5 
6 

8 

10 

ii 

1s 

2O    O 
4 

8 
12 
16 
2020 
24 
28 
32 
36 
2O  4O 
44 
48 
52 
56 
21     O 
4 
8 
12 
16 
2120 
24 
28 
32 
36 
2140 
44 
48 
52 
56 
22    0 
4 
8 
12 
16 
2220 
24 
28 
32 
36 
22  4O 
44 
48 
52 
56 

23    0 
4 
8 
12 
16 
2320 
24 
28 
32 
36 
2340 
44 
48 
52 
56 
24    O 

8.9403 
9417 
9432 
9446 
9460 

8-9475 
9489 

95°3 
9517 
9531 
8-9545 
9559 
9573 
95*7 
9601 

11.0597 

0583 
0568 

0554 
..J?_54£ 
11.0525 
0511 
0497 
0463 
0469 
"•0455 
0441 
0427 

04i3 
0399 

8.9420 
9434 
9449 
94^3 
9477 
8.9492 
9506 
9520 
9534 
9549 
8.9563 
9577 
959i 
9605 
9619 

4 
7 
ii 

4 
7 
ii 

10 

i 

10 

I 

10 

9 

11.0580 
0566 
055  i 
0537 
0523 
11.0508 
0494 
0480 
0466 
045  * 
11.0437 
0423 
0409 

0395 
0381 

10.0017 
0017 
0017 
0017 
0017 

9.9983 
9983 
9983 
9983 
99?3 

9-9983 
9983 
9983 
9983 
9982 

9.9982 
9982 
9982 
9982 
9982 

4O    O 

39  56 
52 
48 
3944 
40 
36 
32 
28 
3924 
20 
16 
12 
8 
39    4 
39    0 
3856 
52 
48 
3844 
40 
36 
32 
28 
3824 
20 
16 
12 
8 
38    4 
38    0 
3756 
52 
48 
3744 
40 
36 
32 
28 
3724 
20 
16 
12 
8 
37    4 
37    0 
3656 
52 
48 
3644 
40 
36 
32 
28 
3624 
20 
16 
12 
8 
36    4 
36    0 

10.0017 
0017 
0017 
0017 
0018 
10.0018 
0018 
0018 
0018 
0018 

8.9614 
9628 
9642 

9655 
9669 

11.0386 
0372 
0358 
°345 
O33i 

8-9633 
9646 
9660 
9674 

9688 

11.0367 

0354 
0340 
0326 
0312 

10.0018 
0018 
0018 
0019 
0019 

9.9982 
9982 
9982 
9981 
9981 

8.9682 
9696 
9709 
9723 
9736 

11.0318 
0304 
0291 
0277 

02b4 

8.9701 
9715 
9729 
9742 
9756 

11.0299 
0285 
0271 
0258 
0244 

10.0019 
0019 
0019 
0019 
0019 

9.9981 
9981 
998i 
9981 
9981 

8.9750 

9763 
9776 
9789 
9803 

11.0250 

0237 

0224 

021  1 
0197 

8.9769 
9782 
9796 
9809 
9823 
8.9836 
9849 
9862 

9875 
9888 

11.0231 
0218 
0204 
0191 
QI77 
11.0164 
0151 
0138 
0125 

0112 

10.0019 
002  o 

OO2O 
0020 
OO2O 

9.9981 
9980 
9980 
9980 
9980 

8.9816 
9829 
9842 

9855 
9868 

11.0184 
OI7I 
0158 

0145 
0132 

IO.OO2O 
OO2O 
0020 
OO2O 
OO2I 

9.9980 
9980 
9980 
9980 
9979 

8.9881 
9894 
9907 
9919 
9932 

II.OII9 
OIO6 
0093 
0081 
0068 
11.0055 
0042 
0030 
0017 
OOO4 

8.9901 

9915 
9927 
9940 
9953 

II.OO99 
0085 
0073 
OObO 
0047 

10.0021 
OO2I 
OO2I 
0021 
OO2I 

9-9979 
9979 
9979 
9979 
9979 

8.9945 
9958 
9970 

9983 
9996 

8.9966 
9979 
9992 
9.0005 
0017 

11.0034 
OO2I 
OOO8 
10.9995 
9983 

IO.OO2I 
OO2I 
0022 
OO22 
OO22 

9-9979 
9979 
9978 
9978 
9978 

9.0008 

0021 

0033 
0046 
0058 

10.9992 

9979 
9967 

9954 
___9942. 
10.9930 
9917 
9905 
9893 
9880 

10.9868 
9856 
9844 
9832 
9820 
10.9808 

9.0030 
0043 

0055 
0068 
0080 

10.9970 

9957 
9945 
9932 
9920 

IO.OO22 
0022 
OO22 
OO22 
0022 

9.9978 
9978 
9978 
9978 
9978 

9.0070 
0083 
0095 
0107 
OI2O 

"9^0132 
0144 
0156 
0168 

_pi8o 
9.0192 

9.0093 
0105 
0118 
0130 
_._OI43 

9-0155 
0167 
0180 
0192 
0204 

10.9907 

9895 
9882 
9870 
9857 
10.9845 

9833 
9820 
9808 
9796 

IO.OO23 
OO23 
0023 
0023 
OO23 
IO.OO23 
OO23 
0023 
OO24 
OO24 

9-9977 
9977 
9977 
9977 
9977 
9-9977 
9977 
9977 
9976 
9976 

9.0216 

10.9784 

IO.OO24 

9.9976 
Sin. 

/ 

m    B 

Cos. 

O'.l     1" 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

ni    & 

Diff. 

Diff. 

95°  =  6h  2O'»  ]                                                                                                     [  5h  36m  = 

84°  . 

TABLE  IX. 


33 


6°  -  Ou  24m  ]             tog:.  Sines  Tangents,  and  Secants.      [  llb  32"'  =  173° 

Difif. 

Diff. 

I 

/ 

111       S 

Sin. 

Cosec 

Tan. 

Cot. 

Sec. 

Cos.  I    m    s 

/ 

O'.l 

1s 

O'.l     1* 

O 

24    0 

9.0192 

10.9808 

9.0216 

10.9784 

10.0024 

9.9976 

36    0 

60 

1 

4 

0204 

i 

3 

9796 

0228 

i 

3 

9772 

0024 

9976 

35  56 

59 

2 

8 

0216 

2 

6 

9784 

0240 

2 

6 

9760 

0024 

9976 

52 

58 

* 

12 

0228 

4 

9 

9772 

0253 

4 

9 

9747 

0024  }     9976 

48  j  57  ; 

4           16 

0240 

5                976° 

0265 

5 

9735 

0024 

9976 

35  44     56 

5     21  20 

9.0252 

6           110.9748 

9.0277 

6            10.9723 

IO.OO2CJ 

9-9975 

40     55  : 

6           24 

0264 

7 

9736 

0289 

7 

9711 

0025  !     9975 

36     54 

7           28 

0276 

8 

9724 

0300 

8 

9700 

0025 

9975 

32 

53 

8            32 

0287 

10 

9713 

0312 

10 

9688 

0025 

9975 

28 

52 

9           36 

0299 

ii 

9701 

0324 

ii 

9676 

0025 

9975 

3524 

51 

10     2440 

9.0311 

110.9689 

9.0336 

10.9664 

10.0025 

9-9975 

20 

50  | 

11 

44 

0323 

i 

3 

9677 

0348 

i 

3 

9652 

0025 

9975 

16 

49 

12 

48 

°334 

2 

6 

9666 

0360 

2 

6 

9640 

0025 

9975 

12 

48  i 

13 

52 

0346 

3 

9 

9654 

0371 

3 

9 

9629 

0026 

9974 

8 

47  i 

14 

56 

°357 

5 

9643 

0383 

5 

9617 

0026 

9974 

35    4 

46 

15 

25    0 

9.0369 

6 

10.9631 

9-0395 

6 

10.9605 

10.0026 

9-9974 

35    0 

45 

16 

0380 

7 

9620 

0407 

7 

9593 

0026 

9974 

3456 

44  i 

17  :        s 

0392 

8 

9608 

0418 

8 

9582 

0026 

9974 

52 

43  ! 

18            12 

0403 

9 

9597 

0430 

9 

9570 

0026 

9974 

48 

42  ! 

19            16 

0415 

10 

9585 

0441 

10 

9559 

0026 

9974 

3444 

41  I 

2O     25  2O 

9.0426 

10.9574 

9-0453 

10.9547 

10.0027 

9-9973 

40 

40 

21           24 

0438 

i 

4 

9562 

0464 

i 

•  3 

9536 

0027 

9973 

36 

39  j 

22           28 

0449 

2 

6 

9551 

0476 

2 

6 

9524 

0027 

9973 

32 

38  i 

23           32 

0460 

3 

9 

9540 

0487 

3 

9 

95  13 

0027 

9973 

28 

37  ! 

24           36 

__°472 

4 

9528 

0499 

5 

9501 

0027 

9973 

3424 

36  i 

25  1  25  40 

9.0483 

5 

10.9517 

9.0510 

6 

10.9490 

10.0027 

9-9973 

2O 

35  i 

26 

44 

0494 

7 

9506 

0521 

7 

9479 

0027 

9973 

16 

34  i 

27 

48 

0505 

8 

9495 

°533 

8 

9467 

0028 

9972 

12 

33 

28 

52 

0510 

9 

9484 

°544 

9 

9456 

0028 

9972 

8 

32 

29 

56 

_?.5i7 

10 

9473 

0555 

10 

9445 

0028 

9972 

34    4 

31 

30 

26    O 

9-0539 

10.9461 

9.0567 

10-9433 

10.0028 

9.9972 

34    0 

30 

31 

4 

0550 

i 

3 

945° 

0578 

1       3 

9422 

0028 

9972 

3356 

29 

32 

8 

0561 

2 

S 

9439 

0589 

2  !     5 

9411 

0028 

9972 

52 

28  ; 

33 

12 

0572 

3 

8 

9428 

0600 

3       8 

9400 

0028 

9972 

48 

27  ; 

34 

16 

_B5«3_ 

4 

9417 

0611 

4   i 

9389 

0029 

997i 

3344 

26 

35 

26  20 

9.0594 

5 

10.9406 

9.0622 

(• 

10.9378 

10.0029 

9.9971 

40 

25 

36 

24 

0605 

7 

9395 

0633 

7 

9367 

0029 

9971 

36 

24  i 

37 

28 

•  0616 

8 

9384 

0645 

8' 

9355 

0029 

9971 

32 

23 

38 

32 

0626 

9 

9374 

0656 

o 

9344 

0029 

9971 

28 

22 

39 

36 

0637 

10 

9363 

0667 

10 

9333 

0029 

9971 

3324 

21 

40 

2640 

9.0648 

10.9352 

'9.0678 

10.9322 

10.0029 

9.9971 

20 

20  i 

41 

44 

0659 

i 

3 

934i 

0688 

I 

3 

9312 

0030 

9970 

16 

19 

42 

48 

0670 

2 

5 

9330 

0699 

2 

9301 

0030 

9970 

12 

18  i 

43 

52 

0680 

3 

8 

9320 

0710 

3 

8 

9290 

0030 

9970 

8 

17  ! 

44 

56 

0691 

4 

93°9 

0721 

4 

9279 

0030 

9970    33    4 

16 

45 

27    0 

9.0702 

5 

110.9298 

9.0732 

5 

10.9268 

10.0030 

9.9970 

33    0 

15 

46 

4 

0712 

7 

9288 

°743 

7 

9257 

0030 

99/0 

3256 

14  i 

47 

8 

0723 

8 

9277 

°754 

8 

9240 

0031 

9969 

52 

13 

48 

12 

0734 

9 

9266 

0764 

9 

9236 

0031 

9969 

48 

12 

49 

16 

0744 

10 

9256 

0775 

10 

9225 

0031 

9969 

3244 

11 

5O 

2720 

9-0755 

10.9245 

9.0786 

10.9214 

10.0031 

9.9969 

40 

10 

51 

24 

0765 

i 

3 

9235 

0796 

i 

3 

9204 

0031 

9969 

36 

9 

52 
53 

28 
32 

0770 
0786 

3 

8 

9224 
9214 

0807 
0818 

2 
S 

i 

9193 
9182 

0031 
0031 

9969 
9969! 

32 

28 

8  ; 
7  ; 

54 

36 

0797 

4 

9203 

0828 

4 

9172 

0032 

9968 

3224 

6 

55 
56 
57 

27  4O 
44 

48 

9.0807 
0818 
0828 

7 

10.9193 
9182 
9172 

9.0839 
0849 
0860 

7 

10.9161 

9J51 
9140 

0.0032 
0032 
0032 

9.9968 
9968 
9968 

20 
16 
12 

5   ! 

4  i 
3 

58 
59 

52 
56 

0838 
0849 

8 
9 

9162 
9I51 

0871 
0881 

8 
9 

9129 
9119 

0032 
0032 

9968 
9968 

8 
32    4 

2  ! 

1  i 

6O 

28    O 

9.0859 

10.9141 

9.0891 

10.9109 

0.0032 

9.9968 

32    0 

0 

O'.l 

1s 

O'.l 

1s 

1 

'    ' 

m     s 

Cos. 

Sec. 

Cot. 

Tan. 

Cosec. 

Sin. 

m    s 

/       1 

DiflF. 

DiflF. 

1 

J  96°  =  6»  24«>  ]                                                                                                      [  5h  89m  =  83°  | 

34 


TABLE  IX. 


|  7°  =  Oh  28'"  ]              L,og.  Sines,  Tangents,  and  Secants.       [  llh 

28»>  -  ] 
m    s 

32    0 
3156 
59 
48 
31  34 
10 
36 
32 
28 
3124 
20 
16 
12 
8 
31    4 
31     0 
3O56 
52 
48 
3044 
40 
36 
32 
28 
3024 
20 
16 
12 
8 
30    4 
3O    O 
2956 
52 
48 
2944 
40 
36 
32 
28 
29  24 
2O 
16 
12 
8 
29    4 
29    0 
2856 
52 
48 
2844 
40 
36 
32 
28 
2824 
20 
16 
12 
8 
28    4 
28    0 

72° 

t 

/ 

m    B 

Sin. 

Diff. 

Cosec 

Tan. 

Diff, 

Cot. 

Sec.      Cos. 

J0'.l 

1« 

O'.l 

1« 

3 
5 
8 

3 

8 

2 

7 

2 

5 
7' 

2 

5 

7 

2 

5 
7 

0 
1 

2 
3 
4 

5 
6 

7 

9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
2O 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

28    O 
4 
§ 
12 
16 
2820 
24 
28 
32 
36 
2840 
44 
48 
52 
56 
29    0 
4 
8 
12 
16 
2920 
24 
28 
32 
36 
2940 
44 
48 
52 
56 
30    0 
4 
8 
12 
16 
3020 
24 
28 
32 
36 
3040 
44 
48 
52 
56 
31    0 
4 
8 
12 
16 
3120 
24 
28 
32 
36 
314O 
44 
48 
52 
56 
32    0 

9.0859 
0869 
0879 
0890 
0900 
9.0910 
0920 
0930 
0940 
0951 
9.0961 
0971 
0981 
0991 

IOOI 

9.1011 
1  020 
1030 
1040 
1050 

i 

2 

3 

4 

5 
6 

8 
9 

i 

2 

3 
4 

6 

8 
9 

i 

2 

3 

4 

I 

I 

9 

i 

2 

3 
4 

3 

3 

5 

7 

2 

S 
7 

2 

5 
7 

2 

5 
7 

2 

5 
7 

10.9141 

9I31 
9121 
9110 
9100 
10.9090 
9080 
9070 
9060 
9049 
10.9039 
9029 
9019 

1078989 

8980 
8970 
8960 
8950 

9.0891 
0902 
0912 
0923 
__°933 
9.0943 

0954 
0964 
0974 
0984 
9.0995 
1005 
1015 
1025 
_™35 
9.1045 

Mil 

1076 

1086 

i 

3 
4 

6 

1 

9 
i 

2 

3 
4 

5 
6 

i 

9 

i 

2 

3 

4 

I 

9 

i 

2 

3 
4 

I 

9 
i 

2 

3 
4 

7 
87 

i 

2 

3 
4 

I 
I 

1  0.9  1  of 
9098 
9088 
9077 
oob  7 
10.9057 
9045 
9036 
9026 
9016 
:  1  0.9005 

8995 
8985 

8975 
8965 

10.8955 

8945 
8934 
8924 
8914 
10.8904 
8894 
8884 
8875 
8865 

10.003 
003 
003 
003 
003 
10.003 

°°3o 
0034 
0034 
0034 
10.0034 
0034 
0034 
°°35 
..._°°35 
10.0035 
0035 
°°35 
0035 
0036 

9.9968 
9967 
9967 
9967 
9967 
9.9967 
9967 
9966 
9966 
9906 
9.9966 
9966 
9966 
9965 
_99j>5 
9.9965 
9965 
9965 
9965 
9964 

60 

59 

58  ; 

57 
56 
55 
54  ! 
53  : 
52 
51 
50 
49 
48 
47 
46 
45 
44  i 
43 
42  i 
41 
40 
39 
38  : 
37  1 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20  j 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 
7 
6 
5 
4 
3 
2 
1 

0 

9.1060 
1070 
1080 
1089 
1099 

10.8940 
8930 
8920 
8911 
8901 

9.1096 
1106 
1116 
1125 
"35 

10.0036 
0036 
0036 
0036 
0036 

9.9964 
9964 
9964 
9964 
9964 

9.1109 
1118 
1128 
1138 
1147 

10.8891 

8872 
8862 
8853 

9-II45 
"55 
1165 

"75 
"85 

10.8855 
8845 

8835 
8825 
8815 

10.0036 
0037 
0037 
0037 
0037 

9.9964 
9963 
9963 
9963 
9963 

9-"57 
1167 
1176 
1186 
"95 

10.8843 

8833 
8824 
8814 
8805 

9.1194 
1204 
1214 
1223 
_i?33 
9.1243 
1252 
1262 
1272 
1281 

10.8806 
8796 
8786 

8777 
8767 

10.0037 
0037 
0038 
0038 
0038 

9.9963 

9963 
9962 
9962 
9962 

9.1205  1      5 
1214]      6 
1224  j      7 

1233        8 
1242  i      9 

10.8795 
8786 
8776 
8767 
8758 

10.8757 
8748 
8738 
8728 
8719 

10.0038 
0038 
0038 
0039 
0039 

9.9962 
9962 
9962 
9961 
9961 
9-996T 
9961 
9961 
9960 
9960 

9.1252 
1261 
1271 
1280 
1289 

i 

2 

3 
4 

7 

I 

i 

2 

3 

4 

! 

i 

10.8748 

8739 
8729 
8720 
8711 
10.8701 
8692 
8683 
8674 
8664 

1^8655 

8646 
8637 
8628 
8619 

9.1291 
1300 
1310 

13*9 
1329 

10.8709 
8700 
8690 
8681 
8671 

10.0039 
0039 
0039 
0040 
0040 

9.1299 
1308 

1317 
1326 

_JL33l 

9.1345 
1354 
1363 
1372 

_J3?i 

9.1390 

1399 
1409 
1418 
-JW 
9.1436 

9.i33f 
1348 
1357 
1367 
1376 

9-1385 
1395 
1404 

Hi3 
1423 

10.8662 
8652 
8643 
8633 
_8624 
10.8615 
8605 
8596 
8587 
8577 

10.0040 
0040 
0040 
0040 
0041 
10.0041 
0041 
0041 
0041 
0041 

9.9960 
9960 
9960 
9960 
_99S9 
9-9959 
9959 
9959 
9959 
9959 

10.8610 
8601 
8591 
8582 

-JS73 
10.8564 

9-1432 
1441 

H5° 
1460 

__L469 
9.1478 

10.8568 
8559 
8550 
8540 

_853i 
10.8522 

10.0042 
0042 
0042 
0042 
0042 
0.0042 

9.9958 
9958 
9958 
9958 
_9958 

9.9958 

/ 

m      B 

Cos. 

O'.l 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

975  =  6u28m]                                                                                                      [5h281"^82° 

TABLE  IX. 


35 


8°  =  Oh  32™  ]              JLogr.  Sines,  Tangents,  and  Secants.       [  llh  24m  =  171° 

'            111      B 

Sill. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     s         ' 

j 

O'.l     1s 

O'.l 

1s 

0 

1 

2 
3 
4 
5 
6 
7 
8 
9 
1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
.33 
34 
35 
36 
37 
38 
39 

40 
41 

42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

32    0 
4 

8 
12 
16 
32  20 
24 
28 
32 
36 
324© 
45 
48 
52 
56 
33    0 
4 
8 
12 
16 
33  2O 
24 
28 
32 
36 
3340 
44 
48 
52 
56 
34    O 
4 
8 
12 
16 
3420 
24 
28 
32 
36 
34  4O 
44 
48 
52 
56 
35    0 
4 
8 
12 
16 
35  2O 
24 
28 
32 
36 
3540 
44 
48 
52 
56 
36    0 

9.1436 
1445 
1453 
1462 
1471 

2 

3 

4 

4 

6 

j 

i 

2 

3 

4 

6 
I 

2 

3 
3 

A 

5 

Q 

I 

2 

3 
3 
4 

I 

6 

7 

i 

2 
2 

4 

1 

6 

7 

i 

2 
2 

3 
4 

6 
6 

7 

2 

s 

7 

2 

5 

7 

2 

6 

2 

I 

2 

4 
6 

2 

4 
6 

10.8564 
8555 
8547 
8538 
_?S29_ 
10.8520 
8511 
8502 

*493 
8484 

9.1478 
1487 
1496 
1505 
I5i5 

i 

2 

3 
4 

5 

8 

i 

2 

3 

4 

5 

I 
I 

i 

2 

3 
4 
4 

8 

i 

2 

3 
3 
4 

8 
i 

2 

3 
3 
4 

I 

6 

7 

i 

2 
2 

3 
4 

6 
6 

7 

2 

5 

7 

2 

5 
7 

2 

5 
7 

2 

I 

2 

4 
6 

2 

I 

10.8522 

8513 
8504 
8495 
8485 
10.8476 
8467 
8458 

8449 
8440 

10.0042 
0043 
0043 
0043 
0043 

9.9958 
9957 
9957 
9957 
"9957 

28    0 
2756 
52 
48 
2744 
40 
36 
'   32 
28 
2724 
20 
16 
12 
8 
27    4 
27    0 
2656 
52 
48 
2644 
40 
36 
32 
28 
2624 
20 
16 
12 
8 
26    4 
26    0 
2556 
52 
48 
2544 
40 
36 
32 
28 
2524 
20 
16 
12 
8 
25    4 
25    0 
2456 
52 
48 
2444 
40 
36 
32 
28 
2424 
20 
16 
12 
8 
24    4 
24    O 

60 
59 
58 
57 
56 
55 
54 
53 
52! 
51 

50 
49 

48 
47 

S 

44! 
43 
42! 
41 
40 
39 
38! 
37  i 
36 
35  j 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
2O  i 
19] 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7! 
6 
5 
4 
3 
2 
1 
0 

9.1480 
1489 
1498 

i5°7 
1516 

9-i525 
1533 
1542 

J551 
1560 

"971568 

1577 
1586 

1594 
1603 

9.1524 

1533 
1542 

*55i 
1560 

10.0043 
0044 
0044 
0044 
0044 

9-9957 
9956 
9956 
9956 
9956 

10.8475 
8467 
8458 
8449 
8440 

9.1569 
1578 
1587 
i596 
1605 

10.8431 
8422 

8413 
8404 

8395 

10.0044 
0044 
0045 
0045 
0045 

9.9956 
9956 
9955 
9955 
9955 

10.8432 

8423 
8414 
8406 
8397 

9.1613 
1622 
1631 
1640 
1649 
9.1658 
1667 

1675 
1684 

1693 

10.8387 
8378 
8369 
8360 
835i 

10.0045 
0045 
0046 
0046 
0046 

9-9955 
9955 
9954 
9954 
9954 

9.1612 
1620 
1629 
1637 
1646 

10.8388 
8380 
8371 
8363 
8354 

10.8342 
8333 
8325 
8316 

8307 

10.0046 
0046 
0046 
0047 
0047 

9-9954 
9954 
9954 
9953 
9953 

9-1655 
1663 
1672 
1680 
1689 

10.8345 

8337 
8328 
8320 
8311 

9.1702 
1710 
1719 

1728 
i736 

10.8298 
8290 
8281 
8272 
8264 

10.0047 
0047 
0047 
0048 
0048 

9-9953 
9953 
9953 
9952 
9952 

9.1697 

1705 
1714 
1722 
i73i 

10.8303 
8295 
8286 
8278 
8269 

9-1745 
1754 
1762 
1771 
1779 

10.8255 
8246 
8238 
8229 
8221 

10.0048 
0048 
0048 
0049 
0049 

9.9952 
9952 
9952 
995  i 
995i 

9-1739 
1747 
i756 
1764 
1772 

10.8261 

8253 
8244 
8236 
8228 

9.1788 
1797 
1805 
1814 
1822 

10.8212 
8203 
8*95 
8186 
8178 

10.0049 
0049 
0049 
0049 
0050 

9-9951 
995  i 
995i 
995  i 
995° 

9.1781. 
1789 
1797 
1806 
1814 

10.8219 
8211 
8203 

8i94 
8186 

9.1831 

1839 
1848 
1856 
1864 

10.8169 
8161 
8152 
8144 
8136 

10.0050 
0050 
0050 
0050 
0051 

9.9950 
995° 
995° 
995° 
9949 

9.1822 
1830 
1838 
1847 
1855 

10.8178 
8170 
8162 
8i53 
8i45 

9-i873 

1881 
1890 
1898 
1906 

10.8127 
8119 
8110 
8102 
8094 

10.0051 
0051 
0051 
0051 
0052 

9-9949 
9949 
9949 
9949 
9948 

9.1863 
1871 
1879 
1887 
1895 

10.8137 
8129 
8121 
8113 
8105 

9-I9I5 
1923 

1931 
1940 
1948 

10.8085 
8077 
8069 
8060 
8052 

10.0052 
0052 
0052 
0052 
°°53 

9.9948 
9948 
9948 
9948 
9947 

9.1903 
1911 
1919 
1927 
T935 

10.8097 
8089 
8081 
8073 
8065 

9.1956 
1964 

1973 
1981 
1989 

10.8044 
8036 
8027 
8019 
Sou 

10.0053 
0053 
0053 
°°53 
0054 

9-9947 
9947 
9947 
9947 
9946 

9-1943 

10.8057 

9.1997 

10.8003 

10.0054 

9.9946 

/ 

m    8 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

98°  =  6h32m]                                                                                                      [5h24m  =  81° 

36 


TABLE  IX. 


9°  =  Oh  36in]               L,og.  Sines,  Tangents  and  Secants.       [llh  2Otn  s  1?O 

/ 

m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

0.0054 
0054 
0054! 
0054 

0055 

Cos. 

m     s 

/    ! 

O'.l 

1s 

O'.l     1* 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
,37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

36    O 
4 
8 
12 
16 
36  2O 
24 
28 
32 
36 
36  4O 
44 
48 
52 
56 
37    0 
4 
8 
12 
16 
3720 
24 
28 
32 
36 
3740 
44 
48 
52 
56 
38    0 
4 
8 
12 
16 
3820 
24 
28 
32 
36 
3840 
44 
48 
52 
56 
39    0 
4 
8 
12 
16 
3920 
24 
28 
32 
36 
3940 
44 
48 
52 
56 
40    0 

9-1943 

195  i 
1959 
1967 

1975 

i 

2 
2 

3 

4 

6 

7 

i 

2 
2 
3 

4 

6 

7 

i 

2 
2 

3 
4 

5 

:| 

7 

2 
2 

3 
4 

5 

7 

i 

2 

2 

3 
4 

4 

'     6 
7 

i 

2 
2 

3 
4 
4 

1 

7 

2 

4 
6 

2 

4 
6 

2 

6 

2 

4 
6 

2 

4 
6 

.  2 

6 

0.8057 
8049 
8041 

8033 
8025 

9.1997 
2005 
2013 

2O22 
2030 

1 

i  i 

2    ' 
2 

3 
4 

6 

7 

2 
2 

3 

4 

6 

7 

2 
2 
3 

4 

5 

7 

i 

2 

2 

3 
4 

5 

6 

7 

2 

2 

3 
4 
4 

i 

7 

i 

2 
2 
3 

4 
4 

I 

7 

2 

4 

6 

2 

4 
6 

2 

4 
6 

2 

6 

2 

4 
6 

2 

I 

10.8003 

7995 
7987 
7978 
7970 

9.9946 
9946 
0946 
9946 
9945 

24    0 
23  56 
52 
48 
2344 
40 
36 
32 
28 
23  24 
2O 
16 
12 
8 
23    4 
23    O 
22  56 
52 
48 
2244 
40 
36 
32 
28 
2224 
20 
16 
12 
8 
22    4 
22    0 
2156 
52 
48 
2144 
40 
36 
32 
28 
2124 
20 
16 
12 
8 
21    4 
21    0 
2O  56 
52 
48 
2O  44 
40 
36 
32 
28 
2O  24 
20 
1O 
12 
8 
20    4 
2O    O 

60! 
59 
•»* 
57 
56 
55 
51 
53 
52 
51 
50 
49 
48 
47 
46 
45  1 
44! 
43  i 
42 
41  I 
40 
39 
38 
37j 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

• 
8 

7 
6 
3 
4 
3i 
9 
1 

0 

9.1983 
1991 
1999 
2007 

201  S 

0.8017 
8009 
8001 
7993 
7985 

9.2038 
2046 
2054 
2062 
2O7O 

10.7962 

7954 
7946 

•7938 
793° 

0.0055 
0055 
°°55 
0055 
0056 

9-9945 
9945 
9945 
9945 
_9944_ 

9-9944 
9944 
9944 
9944 
9943 
9-9943 
9943 
9943 
9943 
9942 

9-2022 
2030 
2038 
2046 
2054 

0.7978 
7970 
7962 

7954 
7946 

9.2078 
2086 
2094 
2102 
2110 

10.7922 

79H 
7906 
7898 
7890 

10.0056 
0056 
0056 
0056 
0057 
10.0057 
0057 
0057 
0057 
0058 

9.2O6l 
2069 
2077 
2085 
2092 

0.7939 

793  i 
7923 
7915 
7908 

9-2II8 
2126 

2134 
2142 
2I5O 

10.7882 

7874 
7866 
7858" 
7850 

9-2IOO 
2108 
2115 
2123 
2I3I 

10.7900 
7892 
7885 
7877 
7869 

9  2158 
2166 
2174 
2181 
2189 

10.7842 

7834 
7826 
7819 
7811 

10.0058 
0058 
0058 
0058 
0059 

9-9942 
9942 
9942 
9942 
9941 

9.2138 
2146 
2153 

2161 
2169 

10.7862 

7854 
7847 
7839 
7831 

9.2197 
2205 
2213 

2221 

2228 

10.7803 
7795 

7787 
7779 
7772 

10.0059 
0059 
0059 
0060 
0060 

9.9941 
9941 
9941 
9940 

_994°_ 
9.9940 
9940 
9940 
9939 
9939 

9.2176 
2184 
2191 

2199 

2206 

10.7824 
7816 
7809 
7801 
7794 

9.2236 
2244 
2252 

2259 
2267 

10.7764 

7756 
7748 
774i 
7733 

10.0060 
0060 
0060 
0061 
0061 

9.2214 

2221 
2229 
2236 
2243 

10.7786 
7779 
7771 
7764 

7757 

9-2275 
2282 
2290 
2298 
2305 

10.7725 
7718 
7710 
7702 
7695 

1  0.006  1 
0061 
0061 
0062 
0062 

9-9939 
9939 
9939 
9938 
9938 

9.2251 

2258 
2266 
2273 
2280 

10.7749 
7742 
7734 
7727 
7720 

9.2313 
232I 
2328 
2336 
2343 

10.7687 
7679 
7672 
7664 
7657 

10.0062 
0062 
0063 
0063 
0063 

9-9938 
9938 
9937 
9937 
9937 

9.2288 
2295 

2303 
2310 
2317 

10.7712 

7705 
7697 
7690 
7683 

9-235I 

2359 
2366 

2374 
2381 

10.7649 
7641 

7634 
7626 
7619 

10.0063 
0063 
0064 
0064 
0064 

9-9937 
9937 
9936 
9936 
9936 
9.9936 
9936 
9935 
9935 
__9935_ 
9-9935 
9934 
9934 
9934 
9934 

9.2324 

2332 
2339 
2346 

2353 

10.7676 
7668 
7661 

7654 
7647 

9.2389 
2396 
2404 
2411 
2419 

10.7611 
7604 
7596 
7589 
758i 

10.0064 
0064 
0065 
0065 
0065 

9.2361 
2368 

2375 
2382 
2390 

10.7639 

7632 
7625 
7618 
7610 

9.2426 
2434 
2441 
2448 
2456 

io-7574 
7566 

7559 
7552 
7544 

10.0065 
0066 
0066 
0066 
0066 

9-2397 

10.7603 

9-2463 

10.7537 

10.0066 

9-9934 

m    s 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1« 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

99°  -  6*  36™  ]                                                                                                      [  5h  20m  =  8O° 

TABLE  IX 


1O°  =  Oh  4Om  ]           Log.  Sines,  Tangents,  and  Secants.      [  llh  16m  =  169- 

.  '     m'  s 

Sin. 

Diff. 

Cosec. 

10.7603 
759^ 
75^9 
7582 
7575 

Tan. 

Di 
0.1 

ff. 
1s 

2 

4 
6 

I 

2 

4 
5 

2 

4 
5 

2 

4 
5 

2 

3 
5 

1s 

Cot. 

Sec. 

Cos. 

m    s        ' 

OM 

i 

2 

3 
4 
4 

I 

6 

i 
i 

2 

3 
4 
4 

I 

6 

2 

3 
4 
4 

6 
i 

2 

3 

4 

4 

6 

i 
i 

2 

3 

••» 
o 

4 
5 

i 
i 

2 

3 

3 

4 
5 

1s 

o 

"M 

2 

A 

4 

6 
7 
8 
9 
1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
i  22 
!  23 
|  24 
25 
!  26 
|  27 
•  28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
4O 
i  41 
i  42 
1  43 
44 
45 
46 
47 
4§ 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

4O    O 
4 

8 
12 
16 
4O2» 
24 
28 
32 
36 
4O4O 
44 
48 
52 
56 
41     0 
4 
8 
12 
16 
4120 
24 
28 
32 
36 
41  4O 
44 
48 
52 
56 
42    0 
4 
8 
12 
16 
42  2O 
24 
28 
32 
36 
4240 
44 
48 
52 
56 
43    O 
4 
8 
12 
16 
43  2O 
24 
28 
32 
36 
43  4O 
44 
48 
52 
56 
44    0 

9.2397 
2404 
2411 
2418 
2425 

2 

4 

5 

2 

5 

2 

4 
5 

2 

4 

5  • 

2 

3 

5 

2 

3 
5 

9.2463 
2471 

2478 
2485 
2493 

i 

2 
2 

3 

4 
4 

7 
i 

2 
2 

3 

4 

4 

7 

2 
2 

3 

4 
4 

6 
6 

i 

2 

3 
4 
4 

I 

6 

i 
i 

2 

3 

4 
4 

3 

6 

i 
i 

2 

3 
4 
4 

6 

10-7537 
7529 
7522 

75i5 
75°7 

10.0066 
0067 
0067 
0067 
0067 

9-9934 
9933 
9933 
9933 
9933 

2O    O 
1956 
52 
48 
1944 
40 
36 
32 
28 
1924 
20 
16 
'  12 
8 
19    4 
19    O 
1856 
52 
48 
1844 
40 
36 
32 
28 
1824 
20 
16 
12 
8 
18    4 
18    0 
1756 
52 
48 
1744 
40 
36 
32 
28 
1724 
20 
16 
12 
8 
1.7    4 
17    O 
1656 
52 
48 
1644 
40 
36 
32 
28 
1624 
2O 
16 
12 
8 
16    4 
16    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
1O 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

9.2432 

2439 

2447 

2454 
2461 

10.7568 
7561 
7553 
7546 
__7539 
IO-7532 
7525 
75i8 

75ii 

7504 

9.2500 
2507 
2515 
2522 
2529 

10.7500 
7493 

7485 
7478 
747i 

10.0068 
0068 
0068 
0068 
0069 

9.9932 

9932 
9932 
9932 

993  i 

9.2468 

2475 
2482 
2489 
2496 

9-2536 
2544 
255i 
2558 
_2565 

9.2573 

2580 

2587 
2594 
2601 

10.7464 
7456 
7449 
7442 
7435 

10.0069 
0069 
0069 
0069 
0070 

9-993  i 
993  i 
993  * 
993  i 
993° 

9-2503 
2510 

2517 
2524 
253i 

10.7497 
7490 
7483 
7476 
7469 

10.7427 
7420 

7413 

7406 

7399 

10.0070 
0070 
0070 
0071 
0071 

9.9930 
993° 
993° 
9929 
9929 

9-2538 
2545 
255i 

2558 

2565 

10.7462 

7455 
7449 
7442 

7435 

9.2609 
2616 
2623 
2630 
2637 

10.7391 
7384 
7377 
7370 
_7363 
10.7356 

7349 
7342 
7334 
7327 

10.0071 
0071 
0071 
0072 
0072 

9.9929 
•  9929 
9929 
9928 
9928 

9-2572 
2579 
2586 

2593 
2600 

10.7428 
7421 

74H 
7407 
7400 

9.2644 
2651 
2658 
2666 
2673 

10.0072 
0072 
0073 
0073 
0073 

9.9928 
9928 
9927 
9927 
9927 

9.2606 
2613 
2620 
2627 
2634 

10.7394 

7387 
7380 

7373 
7366 

9.2680 
2687 
2694 
2701 
2708 

10.7320 

7313 
7306 
7299 
7292 

10.0073 
0074 
0074 
0074 
0074 

9.9927 
9926 
9926 
9926 
9926 

9.2640 
2647 
2654 
2661 
2667 

10.7360 

7353 
7346 
7339 
7333 

9-27I5 
2722 
2729 
2736 
2743 

10.7285 
7278 
7271 
7264 
7257 
10.7250 

7243 
7236 
7230 
7223 

10.0075 
0075 
0075 
0075 
0075 

9.9925 
9925 
9925 
9925 
9925 

9.2674 
2681 
2687 
2694 
2701 

10.7326 
73  19 
7313 
7306 
7299 

9-2750 

2757 
2764 
2770 

2777 

10.0076 
0076 
0076 
0076 
0077 

9.9924 
9924 
9924 
9924 

9923 

9.2707 
2714 
2721 

2727 
2734 

10.7293 
7286 
7279 

7273 
7266 

9.2784 
2791 
2798 
2805 
2812 

10.7216 
7209 
7202 

7195 
7188 

10.0077 
0077 
0077 
0078 
0078 

9.9923 
9923 
9923 
9922 
9922 

9.2740 
2747 
2754 
2760 
2767 

10.7260 

7253 
7246 
7240 
7233 

9.2819 

2825 
2832 

2839 
2846 

10.7181 

7175 
7168 
7161 
7154 

10.0078 
0078 
0079 
0079 
0079 

9.9922 
9922 
9921 
9921 
992  T 

9-2773 
2780 
2786 

2793 
2799 

10.7227 
7220 

7214 
7207 
7201 

9-2853 
2859 
2866 
2873 
2880 

10.7147 
7141 

7134 
7127 
7120 

10.0079 
0080 
0080 
0080 
0080 

9.9921 
9920 
9920 
9920 
9920 

9.2806 

10.7194 

9.2887 

10.7113 

1  0.008  1 

9.9919 

/ 

m     s 

Cos. 

OM     1s 

Sec. 

Cot. 

OM 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

|  1OO°  =  6h  4Om  ]                                                                                                    [  5h  I6m  =  79° 

38 


TABLE  IX. 


11°  =  Oh  44m  ]           L.og.  Sines,  Tangents,  and  Secants.       [  llh  12m  =  168° 

/ 

ni    a 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff, 

Cot. 

Sec.      Cos. 

m    s 

' 

o'.i  !  i«  | 

O'.I     1s 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
1O 
11 

1  12 
13 
14 
15 
!  16 
17 
18 
19 
20 
21 
1  22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
i  55 
56 
57 
58 
59 
60 

44    0 
4 

8 
12 
16 
44  2O 
24 
28 
32 
36 
4440 
44 
48 
52 
56 
45    0 
4 
8 
12 
16 
4520 
24 
28 
32 
36 
4540 
44 
48 
52 
56 
46    0 
4 
8 
12 
16 
4620 
24 
28 
32 
36 
4640 
44 
48 
52 
56 
47    0 
4 
8 
12 
16 
4720 
24 
28 
32 
36 
47  4O 
44 
48 
52 
56 
48    0 

9.2806 
2812 
2819 
2825 
2832 

i 

2 

3 
3 
4 

1 

i 
i 

2 

3 

3 
4 
5 

I 

i 
i 

2 

3 
3 
4 
4 

i 
i 

2 
2 

3 

4 
4 

5 
5 

i 
i 

2 
2 

3 

4 
4 

5 

5 

i 
i 

2 
2 

3 

4 
4 
5 
5 

2 

3 
5 

2 

3 

5 

2 

3 
5 

2 

3 

5 

2 

3 
5 

2 

3 

5 

10.7194 
7188 
7181 

7175 
7168 

9.2887 

2893 
2900 
2907 
2913 

i 

2 

3 
3 
4' 
5 

I 

i 
i 

2 

3 
3 

4 
5 

6 

i 
i 

2 

3 
3 

4 
5 

i 

2 

3 
3 
4 
4 

i 

2 

3 

3 
4 
4 

i 
i 

2 

3 
3 
4 
4 

I 

2 

3 
5 

2 

3 

5 

2 

3 
5 

2 

3 

5 

2 

3 
5 

2 

3 

5 

10.7113 
7107 
7100 

7093 
__7o87 

10.7080 

7073 
7067 
7060 
7053 
10.7047 
7040 

7033 
7027 
7020 

10.0081 
0081 
0081 
0081 
0082 

9.9919 
9919 
9919 
9919 
9918 

16    0 

1556 
52 
48 
1544 
40 
36 
32 
28 
1524 
20 
16 
12 
8 
15    4 
15    0 
14  56 
52 
48 
1444 
40 
36 
32 
28 
11  24 
20 
16 
12 
8 
14    4 
14    O 
1356 
52 
48 
1344 
4O 
36 
32 
28 
1324 
2O 
16 
19 
8 
13    4 
13    0 
1256 
52 
48 
1244 
40 
36 
32 
28 
1224 
20 
16 
12 
8 
12    4 
12    O 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36  ! 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 

i 

7 
6 
9 
4 
3 
2 
1 
0 

9.2838 
2845 
2851 
2858 
_  2864 
9.2870 
2877 
2883 
2890 
2896 

10.7162 
7155 
7M9 
7142 

7i36 
10.7130 
7123 
7117 
7110 
7104 

9.2920 
2927 

2933 
2940 

2947 

9-2953 
2960 
2967 

2973 
2980 

10.0082 
0082 
0082 
0083 
0083 
10.0083 
0083 
0084 
0084 
0084 

9.9918 
9918 
9918 
9917 
9917 
9.9917 
9917 
9916 
9916 
9916 

9.2902 
2909 

2915 
2921 
2928 

10.7098 
7091 
7085 

7°  79 
7072 
10.7066 
7060 

7053 
7047 

7041 
10.7035 

7028 
7022 
7016 
7010 

9.2987 

2993 
3000 
3006 
3013 

10.7013 
7007 
7000 
6994 
6987 

10.0084 
0085 
008  5 
0085 

0085 

9.9916 
9915 
9915 
99!5 
_99£5. 
9.9914 
9914 
9914 
9914 
9913 

9.2934 
2940 
2947 
2953 
2959 

9.3020 
3026 
3°33 
3039 
_3S4A 
9-3052 
3059 
3065 
3072 
3078 

10.6980 
6974 
6967 
6961 
6954 

10.0086 
0086 
0086 
0086 
0087 

10.0087 
0087 
0087 
0088 
0088 

9.2965 
2972 
2978 
2984 
2990 

10.6948 
6941 

6935 
6928 
6922 

9-99I3 
9913 
9913 
9912 

9912 

9.2997 

3°°3 
3009 

3oi5 
3021 

10.7003 
6997 
6991 
6985 
6979 

9-3085 
3091 
3098 

3I04 
3110 

10.6915 
6909 
6902 
6896 
6890 

10.0088 
0088 
0089 
0089 
0089 

9.9912 
9912 
9911 
9911 
9911 

9.3027 

3°34 
3040 
3046 
_3251 
9-3058 
3064 
3070 
3°77 
3083 

I01$ 

6960 
6954 
_694l 
10.6942 
6936 
6930 
6923 
6917 

9-3IJ7 
3^3 
3130 
3'36 
__3J42 

9-3  H9 

3155 
3162 
3168 
3*74 

10.6883 
6877 
6870 
6864 
6858 
10^6851 
6845 
6838 
6832 
6826 

10.0089 
0090 
0090 
0090 
0090 
10.0091 
0091 
0091 
0091 
0092 

9.9911 
9910 
9910 
9910 
_99io 
9.9909 
9909 
9909 
9909 
9908 

9.3089 

3095 
3101 
3107 
_3ii3_ 
9-3IJ9 
3125 
3i3i 
3137 
3143 

10.6911 

9.3181 
3187 

3J93 
3200 
3206 

10.6819 
6813 
6807 
6800 
6794 
10.6788 
6781 

6775 
6769 

_M3 

10.6756 
6750 
6744 

6738 
6731 

10.0092 
0092 
0092 
0093 
0093 

9.9908 
9908 
9908 
9907 
9907 

10.6881 

6875 
6869 
6863 
6857 

9.3212 
3219 
3225 
3231 
3237 

10.0093 
0094 
0094 
0094 
0094 

9.9907 
9906 
9906 
9906 
9906 

9-3I49 
3155 
3161 
3167 
3173 

10.6851 
6845 
6839 

6833 
6827 

9-3244 
325° 
3256 
3262 
3269 

10.0095 
0095 
0095 
0095 
0096 

9.9905 
9905 
9905 
9905 
9904 

9-3I79 

10.6821 

9-3275 

10.6725 

10.0096 

9.9904 

/ 

m      B 

Cos. 

O'.l     I8 

Sec. 

Cot. 

o'.i  i  I- 

Tan. 

Cosec. 

Sin. 

m     B 

/ 

Diff. 

Diff. 

101°  =  6h  44m  ]                                                                                                C  5h  12m  =  78° 

TABLE  IX. 


39 


12°  =  O"  48m  ]            Log.  Sines,  Tangents,  and  Secants.         [II11  «  •  =  167° 

/ 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

m    s 

Sin. 

9-3I79 
31S5 

3i9i 
3*97 

3202 

9.3208 

32H 

3220 
3226 
3232 

Diff. 

Cosec. 

Tan. 

9-3275 
3281 

3287 
3293 
3300 

Diff. 

Cot. 

Sec. 

10.0096 
0096 
0096 
0097 
0097 

Cos. 

9.9904 
9904 
9904 
9903 
9903 

m     s 

/ 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38: 
37 
36 
35 
34 
33 
32 
31 
3O 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 

! 

6 
5 

4: 

2 

1 
O 

O'.l  |  1s 

O'.l 

i 
i 

2 

2 

3 

4 
4 
5 

i 

2 
2 

3 
4 
4 

5 
i 

2 

2 

3 

4 
5 
5 

4 
4 
5 
5 

i 
i 

2 

2 

3 
4 
4 
5 
5 

i 
i 

2 
2 

3 

3 

4 

5 
5 

1s 

48    O 
4 
8 
12 
16 
4821* 
24 
28 
32 
36 
4840 
44 
48 
52 
56 
49    O 
4 
8 
12 
16 
4920 
24 
28 
32 
36 
49  4O 
44 
48 
52 
56 
5O    O 
4 
8 
12 
16 
50  20 
24 
28 
32 
36 
5O4O 
44 
48 
52 
56 
51    0 
4 
8 
12 
16 
5120 
24 
28 
32 
36 
5140 
44 
48 
52 
56 
52    0 

2 
2 

3 
4 
4 
5 

5 

i 

2 
2 

3 

4 
4 

5 
5 

i 

i 

2 
2 

3 

4 

4 
5 
5 

i 
i 

2 

3 
3 
4 
5 
5 

2 
2 

I 
\ 

I 

2 
2 

3 
3 

4 
4 

5 

2 

3 
5 

2 

1 

2 

3 

5 

i 
3 
5 

i 

3 

5 

' 

i 
3 
4 

10.6821 
6815 
6809 
6803 
J5798 
10.6792 
6786 
6780 
6774 
6768 

2 

3 

5 

2 

3 

5 

2 

3 
5 

2 

3 
5 

2 

3 
5 

3 

5 

10.6725 
6719 

6713 
6707 
6700 

12    0 
1156 

52 
48 
1144 
40 
36 
32 
28 
1124 
20 
16 
12 
8 
11    4 
11    0 
1O56 
52 
48 
1044 
40 
36 
32 
28 
1024 
20 
16 
12 
8 
1O    4 
1O    O 
956 
52 
48 
944 
40 
36 
32 
28 
924 
20 
16 
12 
8 
9    4 
9    O 
856 
52 
48 
844 
40 
36 
32 
28 
824 
2O 
16 
12 
8 
8    4 
8    0 

9-3306 
3312 
33i8 
3324 
3330 

10.6694 
6688 
6682 
6676 
6670 

10.0097 
0098 
0098 
0098 
0098 

9.9903 
9902 
9902 
9902 
9962 

9-32  }8~ 
3244 
3250 
3255 
3261 

10.6762 
6756 
6750 
6745 
6  739 

9-3336 
3343 
3349 
3355 
336i 

10.6664 

6657 
6651 
6645 
6639 

10.0099 
0099 
0099 
0099 

OIOO 

9.9901 
9901 
99oi 
9901 

990Q 
9.9900 
9900 
9899 

9899 
9899 

9.3267 
3273 
3279 
3284 
3290 

10.6733 
6727 
6721 
6716 
6710 

9-3367 
3373 
3379 
3385 
339i 

10.6633 
6627 
6621 
6615 
6609 

10.0100 
OIOO 
OIOI 
OIOI 
OIOI 

9.3296 
3302 
3308 
3313 
3319 

10.6704 
6698 
6692 
6687 
6681 

9-3397 
3403 
3409 
34i6 
3422 

10.6603 
6597 
6591 
6584 

6578 

10.0101 
OIO2 
OIO2 
0102 
OIO3 

io.oi6y 

0103 

0103 

0104 
0104 
10.0104 
0104 
0105 
0105 
0105 

9-9899 
9898 
9898 
9898 
9897 

9-3325 
3331 
3336 
3342 
3348 

9-3353 
3359 
3365 
3370 
3376 

10.6675 
6669 
6664 
6658 
6652 
10.6647 
6641 
6635 
6630 
6624 

9-3428 
3434 
3440 
3446 
.._345.2 
9.3458 
3464 
3469 
3475 
348i 

10.6572 
6566 
6560 
6554 
6548 
10.6542 
6536 
6531 
6525 
6^19 

9.9897 
9897 
9897 
9896 

__9M 
9.9896 

9896 
9895 
9895 
•9895 

9-3382 
3387 
3393 
3399 
3404 

10.6618 
6613 
6607 
6601 
6596 

9-3487 
3493 
3499 
3505 
35" 

10.6513 
6507 
6501 

6495 
6489 

10.0106 
0106 
0106 
0106 
0107 

9.9894 

9894 

9894 
9894 
9893 

9.3-410 
34i6 
342i 
3427 
3432 

10.6590 
6584 
6579 
6573 
6568 

9-35  i  7 
3523 
3529 
3535 
354i 

10.6483 

6477 
6471 
6465 
6459 

10.0107 
0107 
0108 

0108 
0108 

9-9893 
9893 
9892 
9892 

_9§92 
9.9892 

9891 
9891 
9891 
9890 
9.9890 
9890 
9890 
9889 
9889 
9.9889 
9888 
9888 
9888 
9888 

9-3438 
3444 
3449 
3455 
3460 

9.3466 
3471 
3477 
3482 
3488 

9-3493 
3499 
35°4 
35io 
3515 

10.6562 
6556 
6551 
6545 
6540 
10.6534 
6529 

6523 
6518 
6512 
10.6507 
6501 
6496 
6490 
6485 

9.3546 
3552 
3558 
3564 
_357o_ 
9-3576 
358i 
3587 
3593 
_3599_ 

9.3605 
3611 
3616 
3622 
3628 

10.6454 
6448 
6442 
6436 
6430 
10.6424 
6419 
6413 
6407 
6401 

10.6395 
6389 
"6384 
6378 
6372 

10.0108 
0109 
0109 
0109 

OIIO 
IO.OIIO 
OIIO 
OIIO 
OIII 
OH  I 
10.01  1  1 
0112 
OII2 
0112 
OII2 

9-3521 

10.6479 

9.3634 

10.6366 

10.0113 

9.9887 

/ 

m    s 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

102°  =  6h  48m  ]                                                                                                      [  5h  8'"  =  77"  : 

40 


TABLE  IX. 


1    : 

13   ._  O':  52m]            Log.  Sines,  Tangents,  and  Secants.         [II1  4m  =  166° 

' 

m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

10.0113 
0113 
0113 
0114 
0114 
10.0114 
0115 
0115 
0115 
0115 

Cos. 

Ill       S 

8     O 
756 
52 
48 
744 
40 
36 
32 
28 
724 
20 
16 
12 
8 
7    4 
7    0 
656 
52 
48 
644 
40 
36 
32 
28 
624 
20 
16 
12 
8 
6    4 
6    0 
556 
52 
48 
544 
4O 
36 
32 
28 
524 
20 
16 
12 
8 
5    4 
5    0 
456 
52 
48 
444 
40 
36 
32 
28 
424 
20 
16 
12 
8 
4 
4    0 

OO 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25  | 
24  i 
23 
22 
21 
2O  i 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3  \ 
2 
J 
O 

O'.l     1s 

O'.l 

1s 

0 

1 

3 
4 
5 

f 

I 
9 
1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
36 
57 
58 
59 
6O 

52    0 
4 
8 
12 
16 
5220 
24 
28 
32 
36 
5240 
44 
48 
52 
56 
53    0 
4 
8 
12 
16 
53  2O 
24 
28 
32 
36 
5340 
44 
48 
52 
56 
51    0 
4 
8 
12 
16 
54  2O 
24 
28 
32 
36 
5440 
44 
48 
52 
56 
55    0 
4 
8 
12 
16 
5520 
24 
28 
32 
36 
5540 
44 
48 
52 
56 
56    0 

9-3521 
3526 
3532 
3537 
3543 
9-3548 
3554 
3559 
35t>4 
357Q 

9-3575 
35^1 
3586 
359i 
3597 
9.3602 
3608 

36i3 
3618 
3624 

i        i 

i       3 

2  \    4 

2 

3 
3  ; 

4  ! 
4  i 

5  ' 

i       i 

i       3 

2  \     4 

2 

3 

3 

4 
4 

5 

i  !     i 

1  i    3 

2    \      4 
2 

3  i 

3 
4  [ 
4 
5 

i       i 

i       3 

2       4 

2 

3 

3 
4 
4 
5 

i       i 

i       3 
2       4 

2 

3 
3 

4 
4 

5 

i       i 

i       3 

2       4 

2 

3 

3  : 

4  i 
4 

10.6479 
6474 
6468 
6463 
..-JJ457. 
10.6452 
6446 
6441 
6436 
6430 

9-3634 
3639 
3645 
3651 
_.3657_ 
9.3662 
3668 

3674 
3680 

3685 

i 
i 

2 
2 

3 

3 
4 
5 
5 

1 

2 
2 

3 
3 

4 

5 

5 

i 

i 

2 
2 

3 
3 

4 
4 

5 

i 

2 
2 

3 

•3 
4 
4 
5 

i 

i 

2 

2 

3 

3 
4 
4 
5 

2 
2 

3 

3 
4 
4 

5 

3 

5 

i 

i 

3 
4 

i 
3 
4 

i 

3 
4 

i 
3 
4 

o7 

10.6366 
6361 
6355 
6349 
6343 
10.6338 
6332 
6326 
6320 
6315 

9.9887 
9887 
9887 
9886 
9886 

9.9886 
9885 
9885 
9885 
9885 
9.9884 
9884 
9884 
9883 
9883 

10.6425 
6419 
6414 
6409 
6403 

9.3691 

3697 
3702 
3708 
37H 

10.6309 
6303 
6298 
6292 
6286 

10.01  16 
0116 
0116 
0117 
0117 

10.6398 
6392 

6387 
6382 
6376 

9-3719 

3725 
373i 
3736 
3742 

10.6281 
6275 
6269 
6264 
6258 

10.0117 
0117 
0118 
0118 
0118 

911| 

9882 
9882 
9882 

9.3629 

3634 
3640 

3645 
3650 

10.6371 
6366 
6360 
6355 
635° 

9-3748 
3753 
3759 
3764 
3770 

10.6252 
6247 
6241 
6236 
6230 

10.0119 
0119 
0119 

OI2O 
0120 

9.9881 
9881 
9881 
9880 
9880 

9-3655 
3661 
3666 
3671 
3677 

10.6345 
6339 
6334 
6329 

6323 
10.6318 

6313 
6308 
6302 
6297 

9.3776 
378i 
3787 
3792 
3798 

10.6224 
6219 
6213 
6208 
6202 

IO.OI2O 
0120 
OI2I 
OI2I 
0121 

9.9880 
9880 
9879 
9879 
9879 

9.3682 
3687 
3692 
3698 
3703 
9.3709 
3713 
3719 
3724 
3729 

9.3804 
3809 

3815 
3820 
3826 

10.6196 
6191 
6185 
6180 
6i74 

10.0122 
OI22 
OI22 
0123 
OI23 

9.9878 
9878 
9878 
9877 
9877 

10.6291 
6287 
6281 
6276 
6271 

9-3831 

3f37 
3842 
3848 
3853 

10.6169 
6163 
6158 
6152 
6147 

10.0123 
OI24 
0124 
0124 
OI24 

9.9877 
9876 
9876 
9876 
9876 

9-3734 
3739 
3745 
375° 
3755 

10.6266 
6261 

6255 
6250 
6245 

9-3859 
3864 
3f7o 
3875 
3881 

10.6141 
6136 
6130 
6125 
6119 

IO.OI25 
0125 
OI25 
OI26 
0126 

9-9875 
9875 
9875 
9874 
9874 

9.3760 

3765 
3770 
3775 
378o 

10.6240 

6235 
6230 
6225 
6220 

9.3886 
3892 
3897 
3903 
3908 

10.6114 
6108 
6103 
6097 
6092 

IO.OI26 
OI27 
0127 
0127 
0128 

9.9874 
9873 
9873 
9873 
9872 

9.3786 
379i 
3796 
3801 
3806 
9.3811 
3816 
3822 
3827 
3832 

10.6214 
6209 
6204 
6199 
._6194 
10.6189 
6184 
6178 

6i73 
6168 

9-39I4 
3919 
3924 
3930 
__3935 
9-3941 
3946 
3952 
3957 
3962 

9.3968 

10.6086 
6081 
6076 
6070 
6065 

10.6059 
6054 
6048 
6043 
6038 

10.0128 
0128 
0128 
0129 
OI29 
IO.OI29 
OI3O 
0130 
OI3O 
OI3I 

9.9872 
9872 
9872 
9871 

_9?_Z]L 
9.9871 
9870 
9870 
9870 

9869 

9.3837  1 

10.6163 

10.6032 
Tan. 

IO.OI3I 

9.9869 

' 

m    B 

Cos. 

O'.l     1" 

Sec. 

Cot. 

O'.l 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

1O3°  =  6h  52m  ]                                                                                                      [  5h  4m  =  76° 

rn 


TABLE  IX. 


41 


;  14° 

=  Oh  56m  ]           Log:.  Sines,  Tangents,  and  Secants.        [  llh  Oni  •-•-  165° 

O 

1 
•» 

3 
4 
5 
6 
7 
8 
9 

i  10 
11 

;  12 
13 
11 
15 
16 
'  B7 
18 
1  19 
20 
21 
22 
23 
I  24  : 
:  25 
26 
27 
:  28 
29 
30 
31 
!  32 
I  33 
34 
35 
36 
!  37 
38 
39 
40 
41 
i  42 
!  43 
44 
45 
46 
i  47 
!  48 
49 
50 
51 
I  52 
!  53 
54 
•55 
!  56 
57 
58 
59 
60 

m     s 

Sin. 

9-3837 
3842 

3847 
3852 
3?57 

9.^862 

3867 
3872 

JI 

9.3887 

3892 

3897 
3902 

3907 
9.3912 

3917 
3922 

3927 
3932 

9-3937 
3942 
3947 
3952 
3957 
9.3961 
3966 
397i 
3976 
398i 
9.3986 
399i 
3996 
4001 
4005 

Di 
OM 

ff. 

i 

3 
4 

i 

3 
4 

3 

4 

i 

3 
4 

i 

2 

4 
i 

2 

4 

Cosec. 

Tan. 

9.3968 

3973 
3978 
3984 
__3?89 

9-3995 
4000 
4005 
4011 
4016 
9.4021 
4027 
4032 

4037 
4042 

Di 
OM 

i 
i 

2 

2 

3 
3 
4 
4 

5 

i 

2 
2 

3 
3 

4 
4 

5 

i 
i 

2 
2 

3 
3 

4 
4 

5 

i 
i 

2 
2 

3 
3 
4 
4 
5 

i 

i 

2 
2 

3 

3 

4 
4 

5 

i 
i 

2 
2 

3 

3 
4 
4 

5 

ff. 
1s 

i 

3 
4 

i 

3 
4 

3 

4 

3 
4 

i 

3 
4 

3 
4 

Cot. 


10.6032 
6027 
6022 
6016 
6011 
10.6005 
6000 
5995 
5989 
5984 
10.5979 

5973 
5968 

5963 
5958 

Sec. 

10.0131 
0131 
0132 
0132 
0132 
10.0133 
OI33 
oi33 
oi33 
oi34 
10.0134 
oi34 
OI35 
oi35 
OI35 

Cos. 

m    s 

/ 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
4O 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

56    0 
4 

8 
12 
16 

56  2t> 
24 
28 
32 
36 
56  4O 
44 
48 
52 
56 
37    0 
4 
8 
12 
16 
5720 
24 
28 
32 
36 
374O 
44 
48 
52 
56 
58    O 
4 
8 
12 
16 
58  20 
24 
23 
32 
36 
584O 
44 
48 
52 
56 
59    O 
4 
8 
12 
16 
59  20 
24 
28 
32 
36 
5940 
44 
48 
52 
56 
60    0 

i 

2 
2 

3 
q 

4 
4 

S 

2 
2 

3 
3 

4 
4 
5 

i 
i 

2 
2 

3 

3 
4 
4 

5 

i 
i 

2 
2 

3 
3 
4 
4 

5 

i 

2 

2 

3 
3 
4 
4 

i 

i 

2 
2 

3 
3 
4 
4 

10.616-^ 
6158 

6i53 
6148 

_6j4_3_ 
10.6138 

6i33 
6128 
6123 
6118 
1  0.6  ii  3 
6108 
6103 
6098 
6093 

9.9869 
9869 
9868 
9868 
9868 

15867 

9867 
9867 
9867 
9866 
9.9866 
9866 
9865 
9865 
9865 

4    0 

356 
52 
48 
344 
40 
36 
32 
28 
324 
20 
16 
12 
8 
3    4 
3    0 
256 
52 
48 
244 
40 
36 
32 
28 
224 
2O 
16 
12 
8 
2    4 
2    0 
156 
52 
48 
144 
40 
36 
32 
28 
124 
2O 
16 
12 
8 
1    4 
1     0 
O56 
52 
48 
044 
40 
36 
32 
28 
O24 
2O 
16 
12 
8 
0    4 
O    0 

10.6088 
6083 
6078 
6073 
6068 

9.4048 

4053 
4058 
4064 
4069 

10.5952 
5947 
5942 
5936 
5931 

10.0136 
0136 
0136 
0137 
0137 

9.9864 
9864 
9864 
0863 
9863 

10.6063 
6058 
6053 
6048 
6043 

9.4074 
4079 
4085 
4090 
4095 

10.5926 
592i 

5915 
59io 

5905 

10.0137 
0138 
0138 
0138 
0139 

9.9863 
9862 
9862 
9862 
9861 

10.6039 
6034 
6029 
6024 
6019 
10.6014 
6009 
6004 
5999 
5995 
10.5990 

5985 
5980 

5975 
__597<> 
10.5965 
596i 
5956 
595  i 
5946 

9.4100 
4106 
4111 
4116 
4121 

"94127" 
4i32 
4137 
4142 

4H7 

10,5900 

5894 
5889 
5884 
_  5879 

10.5873 
5868 

5863 
5858 

5853 

10.0139 
0139 
0140 
0140 
0140 
10.0141 
0141 
0141 
0143 
0142 

9.9861 
9861 
9860 
•9860 
9860 
9.9859 
9859 
9859 
9858 
•9858 
9.9858 
9857 
9857 
9857 
9856 

9.4010 

4015 
4020 
4025 
4030 

94035" 

4039 
4044 
4049 
4054 

94153 
4158 
4163 
4168 
_4J73 
9.4178 
4184 
4189 
4194 
4199 

10.5847 
5842 
5f37 
5832 
_5_82_7 
10.5822 
5816 
5811 
5806 
5801 

10.0142 
0143 
0143 
0143 
0144 

10.0144 
0144 
0145 
0145 
0145 

9.9856 
9856 
9855 
9855 
9855 

9.4059 
4063 
4068 
4073 
_4°_78 
9.4083 
4087 
4092 
4097 
4102 

10.5941 
5937 
5932 
5927 
5922 

9.4204 
4209 
4214 
4220 
4225 

10.5796 

579i 
5786 
578o 
.._577i 
10.5770 

5765 
5760 

5755 
5750 

10.0146 
0146 
0146 
0147 
0147 

9.9854 
9854 
9854 
9853 
9853 

10.5917 

5913 

5908 

5903 
5898 

9.4230 

4235 
4240 

4245 
4250 

10.0147 
0148 
0148 
0148 
0149 

9-9853 
9852 
9852 
9852 
9851 

9.4106 
4111 
4116 
4121 
4125 

10.5894 
5889 
5884 
5879 
5875 

94255 
4260 
4265 
4270 
4275 

io.5745 
5740 
5735 
5730 
5725 

10.0.149 
0149 
0150 
'0150 
0150 

9.9851 

9851 
9850 
9850 
9850 

9.4130 

10.5870 

94281 

10.5719 

10.0151 

9.9849 

/ 

m    s 

Cos. 

OM 

1s 

Sec. 

Cot. 

OM 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

' 

Diff. 

Diff. 

1O4°  =  6h  56'"  ]                                                                                                       [  5h  Om  =  75° 

42 


TABLE  IX. 


15°  =  I     O    ]             Log.  Sines,  Tangents,  and  Secants.       (.  lOh  56m  =  164° 

/ 

in     a 

Sin. 

Dill. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.      Cos. 

1 

m    s 

'    i 

O'.l     I8 

O'.l     I8 

O 

1 
2 
3 

\ 

6 
7 
8 
9 
10 
11 
12 
13 
14 
15 

i  1C 
17 
18 
19 

20 
21 
!  22 
i  23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

0    0 
4 

8 
12 
16 
020 
24 
28 
32 
36 
040 
44 
48 
52 
56 
1     O 
4 
8 
12 
16 
12O 
24 
28 
32 
36 
140 
44 
48 
52 
56 
2    0 
4 
8 
12 
16 
220 
24 
28 
32 
36 
240 
44 
48 
52 
56 
3    0 
4 
8 
12 
16 
320 
24 
28 
32 
36 
34O 
44 
48 
52 
56 
4    0 

9.4130 
4135 
4139 
4144 
4149 

2 
2 

3 
3 
4 
4 

i 

i 
i 

2 

2 

3 

3 
4 
4 

2 
2 

3 
3 
4 
4 

o 

i 
i 

2 
2 
3 

3 
4 
4 

o 

i 
i 

2 
2 
3 

3 
4 
4 

o 

i 
i 

2 
2 

3 
3 
4 
4 

i 

2 

4 

2 

4 

i 

2 

4 
i 

2 

3 

2 

3 

2 

3 

10.5870 
5865 
5861 
5856 
5851 

9.4281 
4286 
4291 
4296 
43°i 

i 

i 

2 
2 

3 
3 
4 

4 

5 

i 
i 

2 
2 

3 
3 

4 

1 

2 
2 

3 

3 
4 
4 

5 

i 

i 

2 
2 

3 
3 
4 

4 
5 

i 
i 
i 

2 

2 

3 
3 
4 
4 

i 

2 
2 
3 

3 
4 
4 

•3 
4 

i 
3 
4 

i 

3 
4 

i 
3 
4 

2 

4 

i 

2 

4 

10.5719 
57H 
5709 
5704 
_5699 
10.5694 
5689 
5684 
5679 
__5^74 
10.5669 
5664 
5659 
5654 
5649 

10.0151 
0151 
0151 
0152 
0152 

9.9849 

9849 
9849 
9848 
9848 

60     0 
59  56 
52 
48 
5944 
40 
36 
32 
28 
59  24 
20 
16 
12 
8 
59    4 
59    O 
5856 
52 
48 
5844 
40 
36 
32 
28 
58  24 
20 
16 
12 
8 
58    4 
58    0 
5756 
52 
48 
5744 
40 
36 
32 
28 
5724 
20 
16 
12 
8 
57    4 
57    O 
56  56 
52 
48 
5644 
40 
36 
32 
28 
56  24 
20 
16 
12 
8 
56    4 
56    O 

6O 
59 

58  ; 

57 

56  : 

55  ! 
54 
53 
52 
51 
50 
49  ! 
48 
17 
46 
45 
44 
43 
42  ! 
41 
40 
39 
38  ! 
37  j 
36 
3-> 
34 
33 

32 
31 
30 
29 
28  i 
27 
26  | 
25 
24 
23 
22 
21 
•JO 
19 
IS 
17 
16 
15 
14 
13 
12 
11 
BO 
9 
8 
7 
6 
5 
4 

a 

2 
1 

O 

/ 

94I53 
4158 
4163 
4168 
4172 

9-4177 
4181 
4186 
4191 
4195 

10.5847 

5842 
5f37 
5832 
5828 

10.5823 

58i9 
5814 
5809 
5805 

9.4306 
43" 
43l6 
4321 
4326 

94331 
4336 
4341 
4346 
4351 

9-4356 
436i 
4366 
437i 
4376 

10.0152 
oi53 

oi53 
oi53 

0154 

10.0154 
OI54 
OI55 
oi55 
oi55 

9.9848 
9847 
9847 
9847 

-Jfc*6 

9.9846 
9846 
9845 
9845 
__9_?45 

9-9844 
9844 
9844 
9843 
9843 

9.4200 
4205 
4209 
4214 
4219 

10.5800 
5795 
579i 
5786 

_57?L 

10.5777 
5772 
5768 

5763 
5758 

10.5644 
5639 
5634 
5629 
5624 

10.0156 
0156 
0156 
oi57 
0157 

9.4223 
4228 
4232 
4237 
4242 
9.4246 
4251 
4255 
4260 
4264 

9.4381 
4386 
4390 

4395 
__4400 

9.4405 
4410 

4415 
4420 

4425 

10.5619 

56l4 
5610 

5605 
5600 

10.0157 
0158 
0158 
0158 
oi59 
10.0159 

0159 
0160 
0160 
0161 

9-9843 
9842 
9842 
9842 
__9l4i 
9.9841 
9841 
9840 
9840 
9839 

10.5754 
5749 
5745 
5740 
5736 

10-5595 

5590 
5585 
558o 

5575 

9.4269 

4274 
4278 

4283 
4287 

io.573i 
5726 
5722 
5717 
_5_7£3 
10.5708 
5704 
5699 
5695 
5690 

10.5686 
5681 

5677 
5672 
•5668 

9.4430 
4435 
4440 
4445 
4449 

10.5570 

5565 
556o 

5555 
555i 

10,0161 
0161 
0162 
0162 
0162 

9.9839 

9839 
9838 

9838 
9838 

9.4292 
4296 
43°  I 
4305 
43  10 

9-4454 
4459 
4464 
4469 
4474 

10.5546 
554i 
5536 
553i 
5526 

10.0163 
0163 
0163 
0164 
0164 

9-9837 
9837 
9837 
9836 
9836 

9.43H 
43  '9 
4323 
4328 
4332 

9-4479 
4484 
4488 

4493 
4498 

10.5521 
55i6 
55i2 
55°7 
5502 

10.0164 
0165 
0165 
0165 
0166 

9.9836 
9835 
9835 
9835 
9834 

9-4337 
434i 
4346 
*  4350 
_^35_5 
9-4359 
4364 
4368 
4372 
4377 
9.4381 
4386 
4390 
4395 
4399 

10.5633 
5659 
5654 
5650 
5645 
10.5641 
5636 

5632 
5628 
.^5613 
10.5619 

56l4 
5610 

5605 
5601 

9-4503 
4508 
4513 
4517 
4522 

10.5497 
5492 
5487 
5483 
5478 

10.0166 
0167 
0167 
0167 
0168 

9.9834 
9833 
9*33 
9833 
9832 

9-4527 
4532 
4537 
4541 
4546 

9-45  5  i 
4556 
456i 
4565 
4570 

io.5473 
5468 

5463 
5459 
_5454 
10.5449 
5444 
5439 
5435 
5450 
10.5425 

10.0168 
0168 
0169 
0169 
0169 
10.0170 
0170 
0170 
0171 
0171 

9.9832 
9832 
9831 
9831 
9831 
9.9830 
9830 
9830 
9829 
9829 
9.9828 

9.4403 

10-5597 

9-4575 

10.0172 

/ 

m     B 

Cos. 

O'.l 

1" 

Sec. 

Cot. 

O'.l  |  I8 

Tan. 

Cosec. 

Sin. 

m     s 

Diff. 

Diff. 

1O5°  =  7h  Om  ]                                                                                                      [  4h  56"1  =  74° 

TABLE  IX. 


16°  =  lh  4M  J              Log.  Sines,  Tangents,  and  Secants.       [  1O'1  52m  =  163° 

/ 

111      S 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     s 

60 
59 
58 
57 
56 
55 
54 
53: 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40! 
39 
38! 
37 
36 
35 
34! 
33 
32  i 
31 
30 
29 
28  ! 
27! 
26 
25 
24 
23 
22 
21 
20 
19 
18: 
17: 
16 
15 
14 
13 
12 
11 
10 
9 
8 

6 

5: 

3! 
2 
1 
0 

O'.l  |  1s 

0.1 

JL- 

i 

2 

4 

2 

4 

i 

2 

4 

i 

2 

4 

i 

2 

3 

2 

3 

17 

O 

1 
2 
3 
4 
5 
6 
7 
8 
9 

1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

4    O 
4 

8 
12 
16 
420 
24 
28 
32 
36 
440 
44 
48 
52 
56 
5    0 
4 
8 
12 

*e 

5  2O 
24 

28 
32 
36 
540 
44 
48 
52 
56 
6    0 
4 
8 
12 
16 
620 
24 
29 
32 
36 
640 
44 
48 
52 
56 
7    O 
4 
8 
12 
16 
720 
24 
28 
32 
36 
74O 
44 
48 
52 
56 
8    0 

9.4403 
4408 
4412 

4417 
4421 

0          I 

I          2 

1       3 

2 

2 

3 
3 
4  i 
4 

o       i 

I          2 

i       3 

2 
2 

3 
3 
4 
4 

0          I 

I          2 

i       3 

2 

2 

3 

3  i 
3 
4  ! 

O           I 
I          2 

i  !    3 

2 
2 

3 

3 

3 

4  1 

i 

O           I 

I          2 

1       3 

2    \ 
2    \ 

3  ; 

3  ! 
3 
4 

oj     , 

!  j 

2    • 

I 

I 

4! 
. 

10-5597 
5592 
5588 

5583 
_5579 

iQ-5575 
5570 
556t> 
5562 
5557 

9-4575 
458o 
4584 
4589 
4594 

i 

2 
2 

3 
3 
4 

4 

i 
i 

i 

2 
2 

3 
3 
4 
4 

i 
i 

i 

2 
2 

3 
3 
4 
4 

i 
i 

i 

2 

2 

3 
3 
4 
4 

o 
i 
i 

2 

2 

3 
3 

4 
4 

o 
i 
i 

2 

2 

3 
3 
4 
4 

10.5425 
5420 
54i6 
54" 
5406 

10.0172 
0172 
0172 

oi73 
0173 

9.9828 
9828 
9828 
9827 
9827 

56    0 
55  56 
52 
48 
55  44 
40 
36 
32 
28 
55  24 
2O 
16 
12 
8 
55    4 
55    0 
5456 
52 
48 
5444 
40 
36 
32 
28 
5424 
20 
16 
12 
8 
54    4 
54    0 
5356 
52 
48 
5344 
40 
36 
32 
28 
53  24 
2O 
16 
12 
8 
53    4 
53    0 
52  56 
52 
48 
5244 
40 
36 
32 
28 
5224 
20 
16 
12 
8 
52    4 
52    0 

9.4425 

443° 
4434 
443s 
4443 

9-4599 
4603 
4608 
4613 
4618 

10.5401 
5397 
5392 
5387 
5382 

10.5378 
5373 
5368 
5363 
5359 

10.0173 
0174 
0174 
0174 
oi75 

9.9827 
9826 
9826 
9826 
9825 

9-4447 
445  2 
445^ 
4400 

4465 

10-5553 
554* 
5544 
5540 
5535 

9.4622 
4627 
4632 
4637 
4641 

10.0175 
0176 
0176 
0176 
0177 

9.9825 
9824 
9824 
9824 
9823 

9.4469 

4473 
4478 
4482 
4486 

io.553i 
5527 
5522 
55i8 
55H 

9.4646 
4651 
4655 
4660 
4665 

10-5354 
5349 
5345 
5340 
5335 

10.0177 
0177 
0178 
0178 
0179 

9.9823 
9823 
9822 
9822 
9821 

9.4491 
4495 
4499 
45°3 
4508 

10.5509 
5505 
55°i 
5497 
5492 

9.4669 
4674 
4679 
4687 
4688 

io.533i 
5326 
532i 
5317 
5312 

10.5307 

5303 
5298 

5293 
5289 

10.0179 
0179 
0180 
0180 
0180 

9.9821 
9821 
9820 
9820 
9820 

9.4512 
45i6 
4521 
4525 
4529 

10.5488 
5484 
5479 
5475 
547i 

9.4693 
4697 
4702 

4707 
4711 

10.0181 
0181 
0182 
0182 
0182 

9.9819 
9819 
9818 
9818 
9818 

9-4533 
453^ 
4542 
4546 
.455° 

10.5467 
5462 
5458 
5454 
5450 

9.4716 
4721 

4725 
473° 
4735 

10.5284 
52/9 
5275 
5270 

5265 

10.0183 
0183 
0183 
0184 
0184 

9.9817 
9817 

9817 
9816 
9816 

9-4555 
4559 
45^3 
4567 
4572 

10.5445 
544i 
5437 
5433 
5428 

94739 
4744 
4748 
4753 
4758 

10.5261 
5256 
5252 
5247 
5242 

10.0185 
0185 
0185 
0180 
0186 

9-9815 
9815 

9815 
9814 
9814 

9-457° 

4580 

4584 

4588 

4593 

10.5424 
5420 
54i6 
5412 
5407 

9.4762 
4767 

477i 

4776 
4781 

10.5238 

5233 
5229 

5224 
5219 

10.0180 
0187 
0187 
0188 
0188 

9.9814 
9813 
9813 
9812 
9812 

94597 
4601 
4605 
4609 
4614 

10.5403 
5399 
5395 
5391 

5386 

9-4785 
4790 

4794 
4799 
4803 

10.5215 
5210 
5206 
5?oi 
5i97 
10.5192 
5187 
5183 
5i78 
5174 

10.0188 
0189 
0189 
0189 
0190 

9.9812 
9811 
9811 
9811 
9810 

9.4618 
4622 
4626 
4630 

4634 

10.5382 
5378 
5374 
5370 
_5366 
10.5361 
5357 
5353 
5349 
5345 

9.4808 
4813 
4817 
4822 
4826 

10.0190 
0191 
0191 
0191 
0192 

9.9810 
9809 
9809 
9809 
9808 

9.4639 

4643 
4647 
4651 
4655 

9-4831 
4835 
4840 

4844 
4849 
9-4853 

10.5169 

5165 
5160 
5156 
5i5i 

10.0192 
0192 

OI93 
0193 
0194 

9.9808 
9808 
9807 
9807 
9806 

9.4659 
Cos. 

10.5341 

10.5147 

10.0194    9-9806 

Toe 

m    s 

O'.l     1s 

Sec. 

Cot. 

O'.l 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

,°  =  7h  4m  ]                                                                                                       [  4h  52m  =  73" 

44 


TABLE  IX. 


17°=  lh  811]               L.og.  Sines,  Tangents,  and  Secants.       [1O11  48m  =  162° 

0 

1 
2 
3 
4 
5 

I 

8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
i  20 
i  21 
22 
23 
24 
25 
26 
27 
28 
29 

30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
'  57 
58 
59 
60 

|    m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

10.0194 
0194 
oi95 
oi95 
0196 

10.0196 
0196 
0197 
0197 
0198 
10.0198 
0198 
0199 
0199 
QI99 

IO.O2OO 
0200 
O2OI 
O2OI 
0201 
IO.O2O2 
O2O2 
0203 
O2O3 
O2O3 

Cos. 

m     s 

52    O 
51  56 
52 
48 
5144 
40 
36 
32 
28 
5124 
2O 
16 
12 
8 
51    4 
51    0 
5O56 
52 
48 
5044 
40 
36 
32 
28 
5024 
2O 
16 
12 
8 
5O    4 
50    0 
4956 
52 
48 
4944 
40 
36 
32 
28 
4924 
2O 
16 
12 
8 
49    4 
49    0 
4856 
52 
48 
4844 
40 
36 
32 
28 
4824 
20 
16 
12 
8 
48    4 
48    0 

i 

O'.l 

1 

O'.l  !  1* 

8    0 
4 
8 
12 
16 
820 
24 
28 
32 
36 
840 
41 
48 
52 
56 
9    0 
4 
8 
12 
16 
9  2O 
24 
28 
32 
36 
94O 
44 
48 
52 
56 
1O    O 
4 
8 
12 
16 
1020 
24 
28 
32 
36 
1040 
44 
48 
52 
56 
11     0 
4 
8 
12 
16 
1120 
21 
28 
32 
36 
1140 
44 
48 
52 
56 
12    0 

9.4659 
4603 
4668 
4672 
4676 

o 

i 

2 

3 
3 
4 

o 
i 

i 

2 

2 
2 

3 
3 
4 

o 
i 

i 

2 
2 
2 

4 

o 

2 

3 
3 

4 

o 
i 

2 
2 
2 

3 
3 
4 

o 

i 

2 
2 
2 

3 
3 
4. 

i 

2 

3 

2 

3 

i 

2 

3 

2 

3 
I 

2 

3 
i 

2 

3 

110.5341 

5337 
5332 
5328 
i  ...S3?4 
10.5320 

53l6 
53i2 
5308 
53°4 
10:5300 
5295 
5291 
5287 
5283 
10.5279 
5275 
5271 
5267 

__5263 
10.5259 
5255 
5251 
5247 
_5243_ 
10.5239 
5235 
5231 
5227 
5223 
10.5219 

5215 
5211 

5207 
5203 

94853 
4858 
4862 
4867 
4871 

o       i 

I          2 

2          3 

2 

3 
3  ; 

4 
4 

o  ,     i 

I          2 

\         3 

2    • 

3 

3 
4 
4 

0    !       I 

I    i      2 

1  ;    3 

2     ! 

2 

3 
3 

4 

4  ; 

O          I 

I          2 

1        3 

2 
2 

3 

3 
4 
4 

O          I 
I         2 

1       3 

2 
2 

3 
3 

4 
4 

0          I 
I          2 

1       3 

2 

2 

3 
3 
4 
4 

.10.5147 

5H2 
5138 
5133 
5L29 
10.5124 
5120 
5H5 
5"i 
5106 

10.5102 
5097 
5093 
5088 

5084 
10.5079 

5075 
5070 
5066 
5061 

IO-5057 
5°53 
5048 

5°44 
5°39 

9.9806 
9806 
9805 
9805 
9804 

60 
59 
58 
57 
56 
.>•> 
54 
53 
52! 
51 
50 
49 
48  i 
47  \ 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32  i 
31 

30 
29 
28: 
27  i 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15! 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

9.4680 
4684 
4688 
4692 
.._46?6_ 
9.4700 

4705 
4709 

4713 
_47i7_ 
9.4721 

4725 
4729 

4733 
4737 

9.4870 
4880 
4885 
4889 

4894 
9.4898 

4903 
4907 
4912 
_49£i 

9.4921 
4925 
493° 
4934 
_4939_ 

9-4943 
4947 
4952 
4956 
4961 

9.9804 
9804 
9803 
9803 
9802 
9.9802 
9802 
9801 
9801 
9801 
9.9800 
9800 
9799 
9799 
9799 

9.4741 
4745 
4749 
4753 
4757 

9.9798 
9798 
9797 
9797 

9797 

9.4761 

4765 
4769 

4773 
4777 

9.4965 
4970 

4974 
4978 

4983 

i°-5°35 
5030 
5026 
5022 
5017 
10.5013 
5008 
5004 
5000 
4995 

IO.O2O4 
0204 
O2O5 
O2O5 
0205 
IO.O2O6 
O2O6 
O2O7 
0207 
O2O7 

9.9796 
9796 
9795 
9795 
9795 

9.4781 

4785 
4789 

4793 
4797 
9.4801 
4805 
4809 

4813 
4817 

9.4987 
4992 
4996 
5000 
5005 

9-9794 
9794 
9793 
9793 

9793 

10.5199 
5195 
5i9i 
5187 
5183 

9.5009 

5°H 
5018 
5022 
5027 

10.4991 
4986 
4982 
4978 
4973 

IO.O2O8 
O2O8 
0209 
O2O9 
O2O9 

9.9792 
9792 
9791 
9791 
9791 

9.4821 
4825 
4829 

4833 
4837 

10.5179 
.     5175 
5i7i 
5l67 
5163 

9-503I 
5035 
5040 

5°44 
5°49 

10.4969 

4965 
4960 

4956 

4951 

IO.O2IO 
0210 
O2  1  1 
O2II 
021  1 

9.9790 
9790 
9789 
9789 
9789 

9.4841 

4845 
4849 

4853 
4857 
9.4861 
4865 
4869 

4873 
4876 

9.4880 
4884 
4888 
4892 
4896 

10.5159 
5J55 
5i5i 
5H7 
5'43 

9-5053 
5057 
5062 
5066 
5070 

10.4947 
4943 
4938 
4934 
493° 

IO.O2I2 
O2I2 
0213 
0213 
O2I3 

9.9788 
9788 
9787 
9787 
9787 

10.5139 
5*35 
5«3« 

5127 
5124 

9-5075 
5079 
5083 
5088 
5092 

10.4925 
4921 

4917 
4912 

4908 
10.4904 
4899 
4895 

10.0214 
0214 
O2I5 
O2I5 
0215 
I0.02I6 
O2l6 
0217 
0217 
0218 

9.9786 
9786 
9785 
97f5 
9785 

10.5120 
5116 

5"2 

5108 
5104 

9.5096 
5101 
5105 
5109 
5"3 

9.9784 
9784 
9783 
97f3 
9782 

9.4900 

10.5100 

9.5118 

10.4882 

10.0218 

9.9782 

r 

, 

m    & 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l     I8 

Tan. 

Cosec. 

Sin. 

m    a 

' 

Diff. 

Diff. 

;iO73  =  7h8m]                                                                                                      [4h48"l  =  72° 

TABLE  IX. 


45 


18=  -  1|J  12'"  ] 

£,og.  Sines,  Tangents,  and  Secants.      [  1O1  44m  =  161° 

O 
I 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

m     s 

12    0 
4 

8 
12 
16 
1220 
24 
28 
32 
36 
124O 
44 
48 
52 
56 
13    0 
4 
8 
12 
16 
13  2O 
24 
28 
32 
36 
1340 
44 
48 
52 
56 
14    O 
4 
8 
12 
16 
1420 
21 
28 
32 
36 
1440 
44 
48 
52 
56 
15    O 
4 
8 
12 
16 
1520 
24 
28 
32 
36 
1540 
44 
48 
52 
56 
16    0 

Sin. 

94900  ! 
49041 
4908 
4911 

49  IS 

Dif 
OM 

F. 
1" 

Cosec. 

Tan. 

9.5118 

5122 
5120 
5Ui 
5135 

Dif 

0.1 

F. 
1" 

Cot. 

Sec. 

0.0218 
0218 
0219 
0219 

O22O 

Cos. 

Ill      S 

,     1 

o 

2 
2 
2 

3 
3 
4 

o 
i 

2 
2 
2 

3 
3 
4 

0 

2 
2 
2 

3 
3 

4 

o 
i 
i 

2 
2 
2 

3 
3 

4 

o 
i 

2 
2 
2 
3 

3 

4 

o 
i 

i 

2 
2 

Tl 

O 

3 
3 

i 

2 

3 

2 

3 
i 

2 

3 

2 

3 

I 

2 

3 

2 

3 

10.5100 
5096 
5092 
5089 
5085 

o 

i  ! 
i 

2 
2 
3 

3 

4 

4 

0 

i 
i 

2 
2 
3 

3 
4 
4 

o 
i 

2 
2 

3 
3 
4 
4 

0 

i 

i 

2 
2 

3 
3 
4 
4 

o 

i 

5 

2 
2 

3 
3 
4 

I 

0 

i 
i 

2 
2 
2 

3 
3 
4 

i  1 

2 

3 

2 

3 
I 

2 

3 

2 

3 

2 

3 

2 

3 

T^ 

10.4882 
4878 

4874 
4869 

4865 

9-9782 
9782 
9781 
9781 
978o 

48    0 
4756 
52 
48 
4744 
40 
36 
32 
28 
4724 
20 
16 
12 
8 
47    4 
47    0 
4656 
52 
48 
4644 
40 
36 
32 
28 
46  24 
20 
16 
12 
8 
46    4 
46    0 
45  56 
52 
48 
4544 
40 
36 
32 
28 
4524 
2O 
16 
12 
8 
45    4 
45    0 
4456 
52 
48 
4444 
40 
36 
32 
28 
4424 
2O 
16 
12 
8 
44    4 
44    0 

60 
59 
58  i 
57 
56 
55 
54 
53 
52 
51 
50  ! 
49  | 
48  ! 
47 
46 
45  1 
44 
43 
42 
41 
40 
39 
38 
37  I 
36 
35 
34 
33  i 
32 
31 
30  i 
29 
28  ! 
27 
26 
25 
24 
23 
22 
21 
20 
19  I 
18 
17  ! 
16 
15  i 
14  ; 
13 
12 
11 
10 
9 
8 
7  ! 
6 
5 
4 
3  \ 
2 
1 
O 

9.4919 

4923 
4927 

493  i 
4935 

10.5081 
5°77 
5073 
5069 
5065 

9-5I39 
5H3 
5H8 
5152 
5156 

10.4861 

4857 

4852 
4848 
4844 

0.0220 
O22O 
O22I 
0221 
O222 

9.9780 
978o 
9779 
9779 
9778 

9-4939 
4942 
4946 

495° 
4954 

10.5061 
5058 
5054 
5050 
5046 

9.5161 
5^5 
5l69 
5i73 
5178 

10.4839 
4835 
4831 
4827 
4822 

IO.O222 
O222 
0223 
O223 
0224 

9-9778 
9778 
9777 
9777 
9776 

9.4958 
4962 

4965 
4969 

4973 

10.5042 
5038 
5°35 
5031 
5027 

9.5182 
5186 
5*90 
5J95 
5199 

10.4818 
4814 
4810 
4805 
4801 

IO.O224 
0225 
O225 
0225 

O226 

9.9776 
9775 
9775 
9775 
9774 

9-4977 
4981 
4984 
4988 
4992 

10.5023 
5019 
5016 
5012 
5008 

9-5203 
5207 
5212 
5216 
5220 

10.4797 

4793 

4788 

4784 
478o 

IO.O226 
0227 
0227 
O227 
0228 

9-9774 
9773 
9773 
9773 
9772 

9.4996 
5000 

5003 
5007 
5011 

10.5004 
5000 
4997 
4993 
4989 

9.5224 
5228 
5233 
5237 
5241 

9-5245 
5249 
5254 
5258 
5262 

10.4776 
4772 
4767 
4763 
4759 

10.0228 
O229 
0229 
0230 
0230 

9.9772 
9771 

977i 
9770 
9770 

9-50I5 
5OI9 
5022 
5026 
5°3° 

10.4985 
498i 
4978 

4974 
4970 

10-4755 

475  i 
4746 

4742 
4738 

10.0230 
023! 
0231 
0232 
0232 

9.9770 
9769 
9769 
9768 
9768 

9.5034 
5°37 
5041 
5045 
'    5°49 

10.4966 
4963 
4959 
4955 
495i 

9.5266 
5270 
5275 
5279 

5283 

10.4734 
473° 
4725 
4721 

4717 

£0.0233 
0233 
0233 
0234 
0234 

9.9767 
9767 
9767 
9766 
9766 

9.5052 
5056 
5060 
5064 
5067 

10.4948 

4944 
4940 

4936 
4933 

9.5287 
5291 
5295 
53°o 
53°4 

10.4713 

4709 

4705 
4700 
4696 

10.0235 

0235 
0236 
0236 
0236 

9-9765 
9765 
9764 
9764 
9764 

9-507I 
5°75 
507^ 
5082 
5086 

10.4929 

4925 
4922 
4918 
49  H 

9-5308 
5312 
53i6 
5320 
5324 

10.4692 
4688 
4684 
4680 
4676 

IO.O237 
0237 
0238 
0238 
0239 

9-9763 
9763 
9762 
9762 
9761 

9.5090 
5093 
5°97 
5101 

5104 

10.4910 
4907 
4903 
4899 
4896 

9-5329 
5333 
5337 
534i 
5345 

10.4671 

4667 
4663 
4659 
4655 

10.0239 
0239 
O24O 
0240 
024! 

9.9761 
9761 
9760 
9760 
9759 

9.5108 
5112 
5"5 
5"9 

5123 

10.4892 
4888 
4885 
4881 
4877 

9-5349 
5353 
5357 
53g 
5366 

10.4651 
4647 

4643 
4638 

4634 

IO.O24I 
O242 
0242 
O242 
0243 

9-9759 
9758 
9758 
9758 
9757 

9.5126 

10.4874 

9-5370 

10.4630 

IO.O243 

9-9757 

/ 

m    s 

Cos. 

OM     1s 

Sec. 

Cot. 

O'.l 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

1O8°  =  7h  12m  ]                                                                                                [  4h  44m  =  71°  | 

46 


TABLE  IX. 


19°  =  I     16™  ]           Log.  Sines,  Tangents,  and  Secants.       [  lOh  4Om  -  16O° 

i        / 

m    & 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     s          '     ; 

O'.l 

I8 

O'.l  |  1» 

O 

1 

2 

3 
4 

5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
I  15 
16 
17 
18 
1  19 
20 
21 
22 
23 
i  24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
i  57 
58 
59 
60 

16    0 

A 

8 
12 
16 
16  2O 
24 
28 
32 
36 
1640 
44 
48 
52 
56 
17    0 
4 
8 
12 
16 
1720 
24 
28 
32 
36 
1740 
44 
48 
52 
56 
18    0 
4 
8 
12 
16 
1820 
24 
28 
32 
36 
1840 
44 
48 
52 
56 
19    0 
4 
8 
12 
16 
1920 
24 
28 
32 
36 
1940 
44 
48 
52 
56 
20    0 

9.5126 
5130 
5134 
5137 
SHi 

9-5  H5 

5H8 
5152 
5i56 
5159 

0 

i 
i 

i 

2 

2 

3 
3 
3 

0 

i 
i 

i 

2 
2 
3 

3 
3 

o 
I 

I 
I 

2 
2 

3 
3 

3 

0 

i 
i 

2 
2 

3 
3 
3 

0 

i 

2 
2 

3 
3 
3 

o 
i 

i 

2 
2 
3 

3 
3 

2 

3 

3 

2 

3 
I 

2 

3 
i 

2 

3 
I 

2 

3 

10.4874 
4870 
4866 
4863 
4859 

9-537° 

5374 
5378 
5382 
5386 

o 
i 
i 

2 
2 
2 

3 
3 
4 

0 

i 

i 

2 
2 
2 
3 

3 

4 

0 

i 
i 

2 

2 
2 

3 
3 
4 

o 

i 
i 

2 
2 

2 

3 
3 

4 

o 

i 
i 

2 
2 
2 
3 

3 
4 

o 

I 
I 

2 
2 
2 
3 

3 
4 

i 

2 

3 

2 

3 
i 

2 

3 

I 

2 

3 
i 

2 

3 

2 

3 

10.4630 
4626 
4622 
4618 
4614 

10.0243 
0244 
0244 
0245 
0245 

9-9757 
9756 

9756 
9755 
9755 

44    0 

4356 
52 
48 
4344 
40 
36 
32 
28 
4324 
20 
16 
12 
8 
43    4 
43    0 
42  56 
52 
48 
4244 
40 
36 
32 
28 
12  24 
2O 
16 
12 
8 
42    4 
42    O 
4156 
52 
48 
4144 
40 
36 
32 
28 
4124 
20 
16 
12 
8 
41    4 
41    0 
4056 
52 
48 
4044 
4O 
36 
32 
28 
4O24 
20 
16 
12 
8 
40    4 
4O    O 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
5O 
49 

48  : 

47  ! 
46 
45  i 
44  i 
43 
42  i 
41 
40 
39 
38 
37 
36  i 
35 
34 
33 
32 
31 
30  ! 
29 
28 
27 
26  i 
25  ! 
24  i 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12  1 
11  1 
1O 
»  i 
8  i 
7 

I 

4 

3 
2 

1 
O 

10.4855 
4852 
4848 
4844 
4841 

9-5390 
5394 
5398 
5402 

_sw. 

9-54" 
5415 
5419 

5423 
5427 

9-543  r 

5435 
5439 
5443 
5447 

10.4610 
4606 
4602 
4598 
4593 

10.0245 
0246 
0246 
0247 
0247 

9-9755 
9754 
9754 
9753 
9753 
9-9752 
9752 
975  J 
9751 
975i 

9-5  163 

5167 
5!7o 
5174 
5*77 
9.5181 
5185 
5188 
5*92 
519^ 

10.4837 

4833 
4830 
4826 
4823 
10.4819 

4815 
4812 
4808 
4804 

10.4589 

4585 
458i 
4577 
4573 

10.0248 
0248 
0249 
0249 
0249 

10.4569 
4565 
456i 
4557 
4553 

10.0250 
0250 
0251 
0251 
0252 

9-975° 
975° 
9749 
9749 
9748 

9-5  199 
5203 
5206 
5210 
5213 

10.4801 
4797 
4794 
4790 

4787 

9-545  i 
5455 
5459 
5463 
5467 

10.4549 
4545 
454i 
4537 
4533 

10.0252 
0253 
0253 
0253 
0254 

9.9748 
9747 
9747 
9747 
9746 

9-52I7 
5221 
5224 
5228 
5231 

10.4783 
4779 
4776 
4772 
4769 

9-5471 

5475 
5479 
5483 
5487 

10.4529 
4525 
452i 
45*7 
4513 

10.0254 
0255 
0255 
0256 
0256 

9.9746 
9745 
9745 
9744 
9744 

9-5235 
5239 
5242 
5246 

5249 

10.4765 
4761 
4758 
4754 
47J5I 

9-5491 
5495 
55oo 
5504 
55o8 

10.4509 

4505 
4500 
4496 
4492 

10.0257 
0257 
0257 
0258 
0258 

9-9743 
9743 
9743 
9742 

9742 

9.5253 
5256 
5260 

5263 
5267 

10.4747 

4744 
4740 

4737 
4733 

9-5512 
55i6 
5520 
5524 
5528 

10.4488 

4484 
4480 
4476 
4472 

10.0259 
0259 
0260 
0260 
0261 

9.9741 
974i 
974° 
9740 

9739 

'  9-5270 
5274 
5278 
5281 
5285 

10.4730 
4726 
4722 
4719 
4715 

9-5531 

5535 
5539 
5543 
5547 

10.4469 

4465 
4461 

4457 
4453 

10.0261 
0261 
0262 
0262 
0263 

9-9739 
9739 
9738 
9738 
9737 

9-5288 
5292 
5295 
5299 
5302 

10.4712 
4708 

4705 
4701 
4698 

9-5551 
5555 
5559 
5563 
5567 

10.4449 

4445 
4441 

4437 
4433 

10.0263 
0264 
0264 
0265 
0265 

9-9737 
9736 
9736 
9735 
9735 

9-5306 
53°9 
5313 
53i6 
5320 

10.4694 

4687 
4684 
4680 

9-5571 

5575 
5579 
5583 
5587 

10.4429 

4425 
4421 

4417 
4413 

10.0266 
0266 
0266 
0267 
0267 

9-9734 
9734 
9734 
9733 
9733 

9-5323 
5327 
5330 
5334 
5337 

10.4677 

4673 
4670 
4666 
4663 
10.4659 

9-5591 
5595 
5599 
5603 
.    5607 
9.5611 

10.4409 

4405 
4401 

4397 
4393 

10.0268 
0268 
0269 
0269 
0270 

9.9732 
9732 
9731 
9731 
973° 

9-5341 

10.4389 

10.0270 

9-9730 

, 
•     / 

m    a 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l  |  I8 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

109°  =  7h  16m  ]                                                                                                 [  4h  4Ora  =  7O7  j 

TABLE  IX. 


47 


2O  J  =a  lb  2O'n  ]            JLog.  Sines,  Tangents,  and  Secants.       [  lOh  36m  =  159° 

O 
1 
2 
3 
4 
5 
6 
7 
8 
9 

i  to 

11 
12 
13 
;  14 
15 
i  16 
17 
1  18 
19 

30 
21 

22 
23 
i  24 
!  25 
126 
27 
i  28 
!  29 
30 
31 
32 
i  33 
i  34 
i  35 
!  36 
j  37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

in     s 

2O     O 
4 
8 
12 
16 
2O  2O 
24 
28 
32 
36 
2O  4O 
44 
48 
52 
56 
21     0 
4 
8 
12 
16 
21  2O 
24 
28 
32 
36 
2140 
44 
48 
52 
56 
22    O 
4 
8 
12 
16 
22  2O 
24 
28 
32 
36 
2240 
44 
48 
52 
56 
23    0 
4 
8 
12 
16 
23  2O 
24 
28 
32 
36 
2340 
44 
48 
52 
56 
24    0 

Sin. 

9-5341 
5344 
5347 
5351 

5354 

9-5358 

536i 
5365 
5368 
5372 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.   '   Cos. 

m     s 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 

48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33  i 
32 
31  1 
30 
29 
28 
27 
26 
251 
24 
23 
22 
21 
20 
19 
18 
17 
16  | 
15  I 
14 
13! 
12 
11 
10 
9! 

s| 

* 

6 

5! 
* 
3 

2 
1 

0 

0.1  i  1* 

O'.l 

o 
i 
i 

2 
2 
2 

3 
3 
4 

o 

2 
2 

2 

3 
3 
4 

o 
i 
i 

2 
2 
2 
3 

3 
4 

o 
I 

2 
2 
2 

3 
3 

4 

o 

I 

2 
2 

1 

4 
o 
i 

2 
2 
2 
3 

3 
4 

1s 

o 
i 
i 

i 

2 

3 
3 
3 

o 

2 
2 

3 
3 
3 

o 
i 
i 
i 

2 

2 

3 
3 
3 

0 

i 

i 
i 

2 
2 
3 

3 
3 

o 

I 

I 

2 
2 
2 

3 
3 

o 
I 
I 

I 

2 
2 

3 
3 

i 

2 

3 

i 

2 

3 

2 

3 

2 

3 

I 

2 

3 
i 

2 

3 

10.4659 
4656 

4653 
4649 
4646 

9.5611 

5615 
56l9 
5622 
5626 

2 
3 

I 

2 

3 

I 

2 

3 
i 

2 

3 
i 

2 

3 
I 

2 

3 

10.4389 
4385 
4381 
4378 
4374 

10.0270  i  9.9730 
0271  i     9729 
0271  !      9729 
0272  i      9728 
0272        9728 

4O    O 

3956 
52 
48 
3944 
40 
36 
32 
28 
3924 
2O 
16 
12 
8 
39    4 
39    O 
3856 
52 
48 
3S44 
40 
36 
32 
28 
3824 
20 
16 
12 
8 
38    4 
38    0 
3756 
52 
48 
3744 
40 
36 
32 
28 
3724 
2O 
16 
12 
8 
37    4 
37    0 
3656 
52 
48 
3644 
40 
36 
32 
28 
3624 
20 
16 
12 
8 
36    4 
36    0 

10.4642 
4639 
4635 
4632 
4628 

9-5630 

$$ 

5642 
5646 

10.4370 
4366 
4362 
4358 
4354 

10.0272 
0273 
0273 
0274 
0274 

9.9728 
9727 
9727 
9726 
9726 

9-5375 
5379 
5382 
53g5 
53^9 

10.4625 
4621 
4618 

4615 
4611 

9-565o 
5654 
5658 
5662 
5665 
9.5669 
5673 
56/7 
5681 
5685 
9.5689 

5693 
5696 
5700 
5704 

10.4350 
4346 
4342 
433s 
4335 

10.0275 
0275 
0276 
0276 
0277 

9.9725 
9725 
9724 
9724 
9723 

9-9723 
9722 
9722 
9722 
9721 
9.9721 
9720 
9720 

97i9 
9719 

9-5392 
5396 
5399 
5402 
54o6_ 
9.5409 

5413 
54i6 
5420 
5423 

10.4608 
4604 
4601 
4598 
4594 
10.4591 
4587 
4584 
458o 
4577 
10.4574 
4570 
4567 
4564 
4560 

1  0-433  ! 

4327 
4323 
43  »9 
4315 
10.4311 
43°7 
43°4 
4300 

4296 
10.4292 

4288 
4284 
4280 
4276 

10.0277 
0278 
0278 
0278 
__0279_ 
10.0279 
0280 
0280 
0281 
0281 

9.5426 
543° 
5433 
5436 
5440 

9.5708 
5712 
57i6 
5720 
5724 

10.0282 
0282 
0283 
0283 
0284 

9.9718 
9718 
9717 
9717 
9716 

9-5443 

5447 
5450 
5453 
5457 

10-4557 
4553 
4550 
4547 
4543 

9-5727 
573i 

5735 
5739 
5743 

10.4273 
4269 
4265 
4261 
4257 

10.0284 
0285 
0285 
0286 
0286 

9.9716 
97i5 
97i5 
97H 
97H 

9.5460 
5463 
5467 
5470 
5474 

10.4540 
4537 
4533 
4530 
4526 

9-5747 
5750 
5754 
5758 
5762 

10.4253 
4250 
4246 
4242 
4238 

10.0286 
0287 
0287 
0288 
0288 

9.9714 
9713 
9713 
9712 
9712 

9-5477 
548o 
5484 
5487 
5490 

10.4523 
4520 
45  i  6. 
4513 
45  10 

9.5766 
5770 
5773 

578i 

10.4234 
4230 
4227 
4223 
4219 

10.0289 
0289 
0290 
0290 
0291 

9.9711 
9711 
9710 
9710 
9709 

9-5494 
5497 
55oo 
5504 
55°7 

10.4506 

4503 
4500 
4496 
4493 

9-5785 
5789 
5792 
5796 
5800 

10.4215 
4211 
4208 
4204 
4200 

10.0291 
0292 
0292 
0293 
0293 

9-9709 
9708 
9708 
9707 
9707 

9-5510 

55H 
5517 
5520 

5523 

10.4490 
4486 

4483 
4480 

4477 

9.5804 
5808 
5811 
5815 
5819 

10.4196 
4192 
4189 
4185 
4181 

10.0294 
0294 
0295 
0295 
0296 

9.9706 
9706 
9705 
9705 
9704 

9.5527 
553° 
5533 
5537 
5540 

10.4473 
4470 
4467 

4463 
4460 

9-5823 
5827 
5830 
5834 
5838 

10.4177 

4173 
4170 
4166 
4162 

10.0296 
0297 
0297 
0298 
0298 

9.9704 
9703 
9703 
9702 
9702 

9-5543 

10.4457 

9.5842 

10.4158 

10.0298 

9.9702 

/ 

m    B 

lo'.i 

Cos.    — 
Di 

1s 

ff. 

Sec. 

Qot. 

0.1 

1s 

Tan. 

Cosec. 

Sin. 

in    s 

/ 

Diff. 

11O°  =  7h  2Om  ]                                                                                                    [  4h  36m  =  69° 

48 


TABLE  IX. 


21°=  I     24m]             Log.  Sines,  Tangents,  and  Secants.      [lOh  32"-  =  158° 

• 

in    a 

Sin. 

Di 
0.1 

Bf. 

Cosec. 

Tan. 

Diff. 

Cot. 

Se::. 

10.0298 
0299 
0299 
0300 

10.0301 
0301 
0302 
0302 
0303 

Cos. 

Ill       S 

36    0 
35  36 
52 
48 
3544 
40 
36 
32 
28 
35  21 
2O 
16 
12 
8 
35     4 
35    O 
34  56 
52 
48 
3444 
10 
36 
32 
28 
3424 
20 
16 
12 
8 
34    4 
34    0 
3356 
52 
48 
3344 
40 
36 
32 
28 
3324 
20 
16 
12 
8 
33    4 
33    0 
32  56 
52 
48 
3244 
40 
36 
32 
28 
32  24 
20 
16 
12 
8 
32     I 
32    0 

6O 
39 
58 
57 
36 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
4.0 
39 
38 
37 
36 
35 
34 
33 
32 
31  i 

3O 
29' 
28 
27 
26 
25 
24 
23 
22 
21! 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11  | 
10  i 
9 
8 
7; 
6 
5 

3 

1 

O 

I8 

O'.l 

1s 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
|55 
56 
57 
58 
59 
60 

24    0 
4 

8 
12 
16 
24  20 
24 
28 
32 
36 
2440 
44 
48 
52 
56 
25    0 
4 
8 
12 
16 
2520 
24 
28 
32 
36 
2540 
44 
48 
52 
56 
26    0 
4 
8 
12 
16 
26  2O 
24 
28 
32 
36 
2640 
44 
48 
52 
56 
27    0 
4 
8 
12 
16 
27  2O 
24 
28 
32 
36 
27  4O 
44 
48 
52 
56 
28    0 

9-5543 

5547 
555° 
5553 
5556 
9-556o 
5563 
5566 
5570 
5573 

0 

i 
i 
i 

2 

2 
2 

3 
3 

o 

i 
i 

i 

2 
2 
2 

3 
3 

0 

i 
i 
i 

2 
2 

2 

3 
3 

o 
i 
i 

i 

2 
2 
2 

3 

o 
i 

2 
2 

2 

3 

3 

o 

i 
i 

i 

2 
2 

2 

3 
3 

i 

2 

3 

i 

2 

3 

2 
2 

I 
2 
2 

I 

2 

2 

2 

2 

io.4457 
4453 
445° 
4447 
4444 

9.5842 
5846 
5849 

5857 

o 
i 

i 

2 
2 

3 
3 
3 
4 

o 

2 
2 

3 
3 
3 
4 

o 
i 

2 

2 

3 
3 

3 

4 

o 

i 
i 

2 
2 

3 
3 
3 

4 

o 

2 
2 
3 

3 
3 

4 

0 

i 

i 

2 

2 

3 
3 
3 
4 

i 

3 

i 

2 

3 

i 

2 

3 
I 

2 

3 
i 

2 

3 

i 

2 

3 

10.4158 
4154 
4I51 
4M7 
4I4J 
10.4139 
4136 
4132 
4128 
4124 

9.9702 
9701 
9701 
9700 
9700 
9.9699 

9699 
9698 
9698 
9697 
9.9697 
9696 
9696 
9695 
._9.69_5. 
9.9694 
9694 
9693 
9693 
9692 

10.4440 
4437 
4434 
443° 
4427 

95861 

5864 
5868 

5872 
5876 

9-5576 
5579 
5583 
5586 

9-5592~ 
5596 
5599 
5602 

5605 

10.4424 

4421 

4417 
4414 

44" 

10.4408 

4404 
4401 

4398 
4395 

9-5879 
5883 
5887 
5891 
5894 

10.4121 
4117 

4"3 

4109 
4106 

10.0303 

0304 
0304 

0305 
°3°5 

9-5898 
5902 
5906 
5909 
5913 

10.4102 
4098 

4094 
4091 

4087 

10.0306 
0306 
0307 
0307 

0308 
10.0308 
0309 

0309 
0310 
0310 
10.0311 
0311 
0312 
0312 
0313 

9.5609 
5612 

5615 
5618 
5621 

'5628 
5631 
5634 
5638 

10.4391 
4388 

4382 
4379 

10.4375 
4372 
4369 
4366 

4362 

95917 
5921 
5924 
5928 

5932 

9-5935 
5939 
5943 
5947 
5950 

10.4083 
4079 
4076 
4072 
4068 

10.4065 
4061 
4057 
4053 
4050 

9.9692 
9691 
9691 
9690 

9.9689 
9689 
9688 
9688 
9687 

9.5641 
5644 
5647 
5650 
5654 

10-4359 
4356 
4353 
4350 
4346 

9-5954 
5958 
596i 
5965 
5969 

10.4046 
4042 
.  4039 
4035 
4031 

10.0313 
°3  H 
03H 
0315 
0315 

9.9687 
9686 
9686 
9685 
9685 

9-5657 
5660 

5663 
5666 
5670 

10-4343 
4340 
4337 
4334 
433° 

9-5972 
5976 
598o 
5984 
5987 

10.4028 
4024 
4020 
4016 
4013 

10.0316 
0316 
0317 

0317 
0318 

9.9684 
9684 
9683 
9683 
9682 

9-5673 
5676 
5679 
5682 

5685 

10.4327 
4324 
4321 

4315 

9-5991 
5995 
5998 
6002 
6006 
9.6009 
6013 
6017 
6020 
6024 

6031 

6035 
6039 
6042 

10.4009 
4005 
4002 
3998 
3994 

10.0318 
0319 
0319 
0320 
0320 

9.9682 
9681 
9681 
9680 
9680 

9.5689 
5692 
5695 
5698 

10.4311 
4308 

43°5 
4302 

4299 

10.3991 

3987 
3983 
398o 
3976 

10.0321 
0321 
0322 
0322 
0323 
10.0323 
0324 
0324 
0325 
0325 

9.9679 

9679 
9678 
9678 
9677 

9-5704 

57" 

57H 
5717 

10.4296 
4292 
4289 
4286 
4283 

10.3972 
3969 
3965 

3958 

9-9677 
9676 
9676 
9675 
9675 

9-5720 

5723 
5726 
5729 
5733 

10.4280 
4277 

4274 
4271 
4267 

9.6046 
6050 
6053 
6057 
6060 

10-3954 
3950 
3947 
3943 
3940 

10.0326 
0326 
0327 
0327 
0328 

9-9674 
9674 
9673 
9673 
9672 

9-5736 

10.4264 

9.6064 

10.3936 

10.0328 

9.9672 

t 

111      B 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

Diff. 

Diff. 

111°  =  7    24-]                                                                                                    [4*32'»:=680| 

TABLE  IX. 


22= 

_  |     2£:i>  i            Log.  Sines,  Tangents,  and  Secants. 

[10" 

Cos. 

9.9672 
9671 
9671 
9670 
9670 
9.9669 
9669 
9668 
9668 
9667 

28n'  =  157° 

i 
in     s 

28    O 
4 
8 
12 
16 
2N  2O 
24 
28 
32 
36 
2840 
44 
48 
52 
56 
29    0 
4 
8 
12 
16 
2920 
24 
28 
32 
36 
2940 
44 
48 
52 
56 
3O    O 
4 
8 
12 
16 
3020 
24 
28 
32 
36 
3O4O 
44 
48 
52 
56 
31    0 
4 
8 
12 
16 
3120 
24 
28 
32 
36 
3140 
44 
48 
52 
56 
32    0 

Sin. 

Di 
0.1 

o 
i 

J 

2 

2 
2 

3 
3 

0 

i 
i 

2 
2 
2 

3 
3 

o 

2 
2 
2 

3 
3 

0 

i 
i 
i 

2 
2 
2 

3 
3 

o 

i 
i 

i 

2 
2 
2 

3 
3 

o 

2 
2 
2 
3 

3 

if. 
1s 

Cosec 

10.4264 
4261 
4258 
4255 
_4252 
10.4249 
4246 
4242 

4239 
4236 

Tan. 

9.6064 
6068 
6071 
6075 
_._6°79 
9.6082 
6086 
6090 
6093 
6097 

Di 
O'.l 

o 

i 
i 

2 
2 

3 
3 
3 

4 

0 

2 

2 

3 
3 
3 
4 

0 

i 
i 

2 

2 

3 
3 
3 

4 

o 
i 

2 
2 

3 
3 
3 
4 

0 

i 

i 

2 

2 

^ 
J 

3 
3 

4 

o 

2 
2 
3 

3 
3 

4 

ff. 
1s 

i 

2 

3 

i 

2 

3 

2 

3 
j 

2 

3 

2 

3 
i 

2 

3 

Cot. 

Sec. 

m    s 

' 

O 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

/ 

9-5736 
5739 

5742 
5745 
_  5748 

9-5751 
5754 
5758 
576i 
5764 

i 

2 

2 

I 
2 
2 

I 

2 

2 

I 
2 
2 

2 

.2 

I 
2 
2 

10.3936 
3932 
3929 
3925 
_  3921 
10.3918 

39H 
3910 

3907 
3903 

10.0328 
0329 
0329 
033° 
__?33<? 
10.0331 

033i 
0332 
0332 
0333 

32    0 
3156 
52 
48 
3144 
4O 
36 
32 
28 
3124 
20 
16 
12 
8 
31    4 
31    O 
3O56 
52 
48 
3O44 
40 
36 
32 
28 
3024 
20 
16 
12 
8 
3O    4 
3O    O 
2956 
52 
48 
2944 
40 
36 
32 
28 
2924 
20 
16 
12 
8 
29    4 
29    0 
2856 
52 
48 
2844 
40 
36 
32 
28 
2824 
20 
16 
12 
8 
28    4 
28    O 

60 
59 

58  : 

57 
56 
55 
54  , 
53 
52  | 
51 
50 
49 
48 
47 
46 
45 
44  | 
43 
42 
41 
40 
39 
38  | 
37 
36 
35 
34 
33  i 
32 
31 
30 
29 
28 
27  ' 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

9.5767 
5770 
5773 
5776 
5779 
9.5782 
5785 
5789 
5792 
5795 

10.4233 
4230 
4227 
4224 
4221 

10.4218 

4215 
4211 

v  4208 
4205 

9.6100 
6104 
6108 
6m 
6115 
~9767i8 
6122 
6126 
6129 
6|33 

10.3900 
3896 
3892 
3889 
3885 
10.3882 
3878 
3f74 
3871 
3867 

10.0333 
°334 
°334 
°335 
Q336 
10.0336 
0337 
0337 
0338 
0338 

9.9667 
9666 
9666 
9665 
_9664 
9.9664 
9663 
9663 
9662 
9662 

9-5798 
5801 
5804 

5807 
5810 

10.4202 
4199 
4196 
4193 
4190 
10.4187 
4184 
4181 
4178 
4i/5 
10.4172 
4169 
4166 
4162 
4159 
10.4156 
4153 
4I50 
4H7 
4144 

9.6136 
6140 
6144 
6147 
6151 

9.6154 
6158 
6162 
6165 
6169 
9.6172 
6176 
6179 
6183 
6187 
9.6190 
6194 
6197 
6201 
6204 

10.3864 
3860 
3856 
3853 
__3l49 
10.3846 
3842 
3838 
3835 
3831 
10.3828 

3824 
3821 

38i7 
_!?!3 
10.3810 
3806 
3803 
3799 
3796 

10.0339 

°339 
0340 
0340 
034i 
10.0341 
0342 
0342 
°343 
__°343 
10.0344 
0344 
°345 
°345 
0346 
10.0346 

°347 
0348 
0348 
0349 

9.9661 
9661 
9660 
9660 
_9j>59 
9.9659 
9658 
9658 
9657 
^965  7 
9.9656 
9656 
9655 
9655 
_9_654 
9.9654 

9653 
9652 
9652 
9651 

9-58i3 
5816 

5819 

5822 

5825 
9.5828 
5831 
5g34 
5838 
_5?4i 
9.5844 
5847 
5850 

5853 
5856 

9-5859 
5862 

5865 
5868 
5871 

10.4141 
4138 
4135 
4I32 
4129 

9.6208 
6211 
6215 
6219 
6222 

10.3792 
3789 
3785 
378i 
3778 

10.0349 
°35° 
035° 
0351 
035i 

9.9651 
9650 
9650 
9649 
9649 

9-5874 

5877 
5880 

5883 
5886 

10.4126 
4123 
4120 
4117 
4114 

9.6226 
6229 

6233 
6236 
6240 

io-3774 
3771 
3767 
3764 
3760 

10.0352 
0352 
0353 
0353 
°354 

9.9648 
9648 
9647 
9647 
9646 

9.5889 
5892 
5895 
5898 
5901 

10.4111 

4108 
4105 
4102 
4099 

9.6243 
6247 
6250 
6254 
6257 

iQ-3757 
3753 
3750 
3746 
•  3743 

10.0354 
°355 
0355 
0356 
0357 

9.9646 
9645 
9645 
9644 

9643 

9.5904 

5907 
59io 

5913 
59i6 

10.4096 

4093 
4090 

4087 
4084 

9.6261 
6264 
6268 
6271 
6275 

10.3739 
3736 
3732 
3729 
3725 

10.0357 
0358 
0358 
0359 
°359 

9.9643 
9642 
9642 
9641 
9641 

9-59I9 

10.4081 

9.6279 

10.3721 

0.0360 

9.9640 

m    s 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

1 

Diflf. 

Diff. 

112°  =  7h  28ra  ]                                                                                                    [  411  28m  —  67° 

50 


TABLE  IX. 


23°  =  lh  32"'] 

Log.  Sim-*,  Tangents,  and  Secants.       [  lo    •*!     -  156° 

/ 

m    s 

Sin. 

Di 
OM 

S. 
1 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m    s 

28    0 
2756 
52 
48 
2744 
40 
36 
32 
28 
27  24 
20 
16 
12 
8 
27    4 
27    0 
26  56 
52 
48 
2644 
40 
36 
32 
28 
2624 
2O 
16 
12 
8 
26    4 
26    O 
2556 
52 
48 
2544 
40 
36 
32 
28 
25  24 
20 
16 
12 
8 
25    4 
25    O 
2456 
52 
48 
2444 
40 
36 
32 
28 
2424 
20 
16 
12 
8 
24    4 
24    0 

6O 
39 
58 
57 
56 
55 
54 
5:i 
52 
51 
50 
49  ! 
48 
47  i 
46  \ 
45 
44 
43  ; 
42 
41 
40 
39 
38  j 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18  j 
17  ! 
16 
15  ! 
14  i 
13  1 
12 
01 

10 

9 

8 

6 
5 
4 

3 
2 

1 
0 

O'.l     1 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
!  35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

32    0 
4 
8 
12 
16 
32  20 
24 
28 
32 
36 
3240 
44 
48 
52 
56 
33    0 
4 
8 
12 
16 
3  8  2O 
24 
28 
32 
36 
3340 
44 
48 
52 
56 
34    0 
4 
8 
12 
16 
34  2O 
24 
28 
32 
36 
3440 
44 
48 
52 
56 
35    0 
4 
8 
12 
16 
3520 
24 
28 
32 
36 
3540 
44 
48 
52 
56 
36    O 

9-59I9 
5922 

5925 
592« 
5931 

0 

i 

i 
i 

2 
2 
2 

3 

3 

0 

i 

i 

2 

3 
3 

0 

i 
i 
i 

2 
2 
2 
3 

3 

o 
I 

2 

3 
3 

o 
i 

i 
i 

2 
2 
2 
3 

3 

o 

I 
I 

I 

2 
2 

2 

3 
3 

i 

2 
2 

2 

2 

I 

2 
2 

I 

2 
2 

2 
2 

2 
2 

10.4081 
4078 

4°  75 
4072 
4069 

9.6279 
6282 
6286 
6289 
6293 

6      i 

I          2 

1       3 

: 

3 
3  1 
3 
4  j 

0          I 
I          2 

2          3 

o 

3 
3 
3 
4 

O          I 
I         2 

1       3 

2 

2   1 

3 
3 

3  ! 
4  ; 

O          I 
I          2 

1       3 

2 
2 

3 
3 
3 
4 

0          I 
I          2 

i       3 

2 
2 

3 
3 

3  ! 

0          I 
I          2 

i       3 

2 

2 

3 
•  3 
3 
4 

10.3721 
37i8 
37H 
37" 

3707 

10.0360 
0360 
0361 
0361 

0362 
IO.O362 
0363 
0364 
0364 

0365 

9.9640 
9640 
9639 
9639 
9638 

9.9638 
9637 
9636 
9636 

9635 

9-5934 
5937 
5940 
5943 
5945 

10.4066 
4063 
4060 
4057 
4055 

9.6296 
6300 

6303 
6307 
6310 
9.6314 

6317 
6321 
6324 
6328 

10.3704 
3700 
3697 
3693 
3690 

9.5948 
595i 
5954 
5957 
5960 

10.4052 
4049 
4046 

4043 
4040 

10.3686 
3683 
3679 
3676 
3672 

10.0365 
0366 
0366 
0367 
0367 

9-9635 
9634 
9634 
9633 
__.9633 
9.9632 
9632 
9631 
9631 
9630 
9.9629 
9629 
9628 
9628 
9627 

9-5963 
5966 

5969 
5972 
_5975 
9-59/8 
598i 
5984 
5987 
5990 

10.4037 
4034 
4031 
4028 
4025 

9-633I 
6334 
6^38 
6341 
6345 

10.3669 
3666 
3662 
3659 

3^55 

10.0368 
0368 
0369 
0369 
03/0 

10.4022 
4019 
4016 

4013 
4010 

9.6348 
6352 
6355 
6359 
6362 

10.3652 
3648 
3645 
3641 
_36J8 
10.3634 
3631 
3627 
3624 
3620 

IO.O37I 

0371 
0372 
0372 
0373 

9-5992 
5995 
5998 
6001 
6004 

10.4008 
4005 
4002 
3999 
3996 

9.6366 
6369 

6373 
6376 
6380 

10.0373 
0374 
0374 
°375 

_°175 
10.0376 
0377 
0377 
0378 
0378 

9.9627 
9626 
9626 
9625 
9625 

9.6007 
6010 
6013 
6016 
6019 

9.6021 
6024 
6027 
6030 

__6°33 
9.6036 
6039 
6042 
6045 
6047 

10.3993 
3990 
3987 
3984 
398i 

9-6383 
6386 
6390 
6393 
6397 
9.6400 
6404 
6407 
6411 
6414 
9.6417 
6421 

6424 
6428 

6431 

10.3617 

3614 
.  3610 
3607 
3603 

9.9624 
9623 
9623 
9622 
9622 

10.3979 
3976 
3973 
3970 
3967 
10.3964 
3961 
3958 
3955 
3953 

10.3600 
3596 
3593 
3589 
3586 

10-3583 
3579 
3576 
3572 
3569 

10.0379 

0379 
0380 
0380 
0381 
10.0382 
0382 
0383 
0383 
0384 

9.9621 
9621 
9620 
9620 
__?6l9 
9.9618 
9618 
9617 
9617 
9616 
9.9616 
9615 
9615 
9614 

9613 
9.9613 
9612 
9612 
961  1 
96  n 
9.9610 
9610 
9609 
9608 
9608 

9.6050 

?5? 
6056 

6059 
6062 
9.6065 
6068 
6070 
6073 
6076 
0.6079 
6082 
6085 
6087 
6090 

10.3950 
3947 
3944 
394i 
.3938 

10-3935 
3932 
3930 
3927 
3924 
10.3921 
39i8 
3915 
3913 
3910 

9-6435 
6438 
6441 

6445 
6448 

9.6452 
6455 
6459 
6462 
_  6465 
9.6469 
6472 
6476 
6479 
6482 

10.3565 
3562 
3559 
3555 
__3552 
10.3548 
3545 
354J 
3538 
3535 

i°-353  i 
3528 
3524 
352i 
35i8 

10.0384 
0385 
0385 
0386 

_°387 
10.0387 
0388 
0388 
0389 
_53?9 
10.0390 
0390 
0391 
0392 
0392 

9.6093 

10.3907 

9.6486 

10.3514 

10.0393 

9.9607 

/ 

m    s 

Cos. 

O'.l     1* 

Sec. 

Cot. 

O'.l     1s 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

113°^7h32m]                                                                                                    [4h24m  =  66° 

TABLE  IX. 


51 


24°  =  lh  36rn  ]            Log.  Sines,  Tangents,  and  Secants.      [  lOh  2Om  -  155° 

'        m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m    s 

i 

O'.l 

I8 

O'.l 

I8 

O 

1 
2 
3 
4 

U 

8 
9 

i!  10 
11 
!  12 

!  13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
6® 

36    O 
4 
8 
12 
16 
36  2O 
24 
28 
32 
36 
36  4O 
44 
48 
52 
56 
37    0 
4 
8 
12 
16 
37  2O 
24 
28 
32 
36 
3740 
44 
48 
52 
56 
38    0 
4 
8 
12 
16 
38  2O 
24 
28 
32 
36 
3840 
44 
48 
52 
56 
39    0 
4 
8 
12 
16 
3920 
24 
28 
32 
36 
39  4O 
44 
48 
52 
56 
4O    O 

9.6093 

6096 
6099 
6102 
6104 

o 

i 
i 
i 

2 
2 
2 
2 

O 

I 
I 
I 
I 
2 
2 
2 
2 

0 
I 

I 
I 
2 
2 
2 
2 

O 

I 

I 

I 
2 
2 
2 
2 

0 

2 
2 
2 
2 

O 
I 

I 
I 
I 
2 
2 
2 
2 

2 

2 

\ 

2 
2 

I 

2 

2 

2 
2 

I 

2 
2 

10.3907 

39°4 
3901 

3898 
3896 

9.6486 
6489 

6493 
6496 
6499 

o 

i 
i 

2 
2 
3 

3 
3 

4 

0 

i 

i 

2 
2 

3 

3 
3 
4 

o 

2 
2 
3 

3 

3 

4 

o 

i 
i 

2' 
2 
3 

3 

3 
4 

0 

i 

2 
2 

3 
3 
3 
4 

0 

i 

i 

2 
2 
3 

3 
3 

4 

i 

2 

3. 

I 

2 

3 

i 

2 

3 
i 

2 

3 

I 

2 

3 
i 

2 

3 

10.3514 
35" 
35°7 
3504 
35oi 

0.0393  j  9.9607 
0393       9607 
0394  !      9606 
0394       9606 
0395       9605 

24    O 

2356 
52 
48 
2344 
40 
36 
32 
28 
2324 
20 
16 
12 
8 
23    4 
23    O 
2256 
52 
48 
2244 
40 
36 
32 
28 
2224 
20 
16 
12 
8 
22    4 
22    0 
2156 
52 
48 
2144 
40 
36 
32 
28 
2124 
2O 
16 
12 
8 
21    4 
21    0 
2056 
52 
48 
2044 
40 
36 
32 
28 
2024 
20 
16 
12 
8 
20    4 
2O    O 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

9.6107 
6110 
6113 
6116 
6119 
9.6121 
6124 
6127 
6130 
6133 

10.3893 
3890 
3887 
3884 
3881 

9-6503 
6506 
6510 

6516 

10.3497 
3494 
3490 
3487 
3484 
10.3480 
3477 
3473 
3470 
3467 

0.0396 
0396 
0397 
0397 
0398! 

9.9604 
9604 
9603 
9603 
9602 
"9^9602 
9601 
9601 
9600 
9599 
9-9599 
9598 
9598 
9597 
9597 

10.3879 
3876 

3873 
3870 

3867 

9.6520 

6523 
6527 

6530 
6533 

10.0398 
0399 
0399 
0400 
0401 

9-6I35 
6138 
6141 
6144 

10.3865 
3862 
3859 

9-6537 
6540 

6543 
6547 
6550 

9-6553 
6557 
6560 
6564 
6567 
9.6570 
6574 
6577 
6580 
6584 

10.3463 
3460 
3457 
3453 
3450 

10.0401 
0402 
0402 
0403 
0403 

9.6149 
6152 

6158 
6161 
9.6163 
6166 
6169 
6172 
6174 

10.3851 
3848 

3845 
3842 

3839 

10.3447 

3443 
3440 
3436 
3433 

10.0404 

0405 
0405 
0406 
0406 

9.9596 
9595 
9595 
9594 
9594 

10.3837 
3834 

3831 
3828 

10.3823 
3820 

3814 
3812 

10.3430 
3426 

3423 
3420 

10.0407 

0407 
0408 
0409 
0409 

9-9593 
9593 
9592 
9591 
9591 

9.6177 
6180 
6183 
6186 
6188 

9.6587 
6590 
6594 
6597 
6600 

10.3413 

3406 

3403 
3400 

10.0410 
0410 
0411 
0411 
0412 

9.9590 
9590 
9589 
9589 
9588 

9.6191 
6194 

6i97 
6199 
6202 

10.3809 
3806 
3803 
3801 
3798 

9.6604 
6607 
6610 
6614 
6617 

10.3396 
3393 
339° 
3386 
3383 
10.3380 
3376 
3373 
3370 
3366 

10.0413 

0413 
0414 
0414 
0415 

9.9587 
9587 
9586 
9586 
9585 

9.6205 
6208 
6210 
6213 
6216 

10-3795 
3792 
3790 
3787 
3784 
10.3781 
3779 
3776 
3773 
3770 
10.3768 

3765 
3762 
3760 
3757 

9.6620 
6624 
6627 
6630 
6634 

10.0416 
0416 
0417 
0417 
0418 

9-9584 
9584 
9583 
9583 
9582 

9.6219 
6221 
6224 
6227 
6230 
9.6232 
6235 
6238 
6240 
6243 

9.6637 
6640 
6644 
6647 
6650 

10.3363 

3356 
3353 
_335?_ 
10.3346 

3343 
3340 
3336 
3333 

10.0418 
0419 
0420 
0420 
0421 

9.9582 

958o 
9579 

9.6654 
6657 
6660 
6664 
6667 

10.0421 
0422 
0423 
0423 
0424 

9-9579 
9578 
9577 
9577 
9576 

9.6246 
6249 
6251 
6254 
6257 

10-3754 
375i 
3749 
3746 
3743 

9.6670 
6674 
6677 
6680 
6683 

10.3330 
3326 

3323 
3320 

10.0424 
0425 
0425 
0426 
0427 

9-9576 

9575 
9575 
9574 
9573 
9-9573 

9.6259 

10.3741 

9.6687 

10.3313 

10.0427 

L 

m    B 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

Tan. 

Cosec. 

Sin. 

m    s 

' 

Diff. 

Diff. 

|  114°  =  7h  36m  ]                                                                                                    [  4h  2Ora  =  65° 

52 


TABLE  IX. 


25°  =  1"  4O'n  ] 

JLog.  Sines,  Tangents,  and  Secants.       [  1O1'  16'"  =  1 

51 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 
33 
32 
31 

3O 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
1O 
9 
8 
7 
6 
5 
4 
3 
9 
1 
0 

m    s 

Sin. 

Diff. 
0.1     1^ 

Cosec. 

Tan. 

9.6687 
6690 
6693 
6697 
6700 

Di 
0.1 

0 

2 
2 

3 
3 
3 

0 

i 

i 
i 

2 
2 

3 
3 
3 

0 

i 

i 
i 

2 

2 

3 
3 
3 

0 

i 

i 

2 
2 

3 
3 
3 

0 

i 
i 
i 

2 
2 
3 

3 
3 

0 

I 
I 

I 

2 

2 

3 
3 
3 

TvT 

0f. 

1- 

i 

2 
2 

I 

2 
2 

2 

2 

2 

2 

2 

2 

I 
2 
2 

Cot. 

io.33'3 
33i° 
3307 
3303 
33°° 

Sec. 

Cos. 

9-9573 
9572 
9572 
957i 
9570 

in     s 

20     O 
1956 
52 
48 
1944 
40 
36 
32 
28 
1924 
20 
16 
12 
8 
19    4 
19    0 
1856 
52 
48 
1844 
40 
36 
32 
28 
1824 
2O 
16 
12 
8 
18    4 
18    0 
1756 
52 
48 
1744 
40 
36 
32 
28 
1724 
20 
16 
12 
8 
17    4 
17    0 
1656 
52 
48 
1644 
40 
36 
32 
28 
1624 
20 
16 
12 
8 
16    4 
16    O 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 

i  ri 
15 
!  16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
6O 

40    0 
4 
8 
12 
16 
4020 
24 
28 
32 
36 
4O4O 
44 
48 
52 
56 
41    0 
4 
8 
12 
16 
4120 
24 
28 
32 
36 
41  4O 
44 
48 
52 
56 
42    0 
4 
8 
12 
16 
4220 
24 
28 
32 
36 
42  4O 
44 
48 
52 
56 
43    0 
4 
8 
12 
16 
4320 
24 
28 
32 
36 
4340 
44 
48 
52 
56 
44    0 

9.6259 
6262 
6265 
6268 
6270 

0 

i 

i 
i 

2 

2 
2 

2 

O 

I 
I 
I 
I 

2 

2 
2 

2 

O 

I 

2 

2 
2 

2 

0 
I 
I 
I 

I 
2 
2 
2 
2 

0 
I 

I 
I 
I 
2 
2 
2 
2 

O 
I 
I 
I 
I 
2 
2 
2 
2 

• 

i 

2 
2 

1 
2 
2 

I 

2 

2 

I 
2 
2 

I 

2 
2 

10.3741 
3738 
3735 
3732 
3730 

10.0427 
0428 
0428 
0429 
0430 

9.6273 
6276 
6278 
6281 
6284 

10.3727 

3724 
3722 

3719 
37i6 

9.6703 
6706 
6710 

J7I3 
6716 

10.3297 

3294 
3290 
3287 
3284 

10.0430 

°43  * 
043  I 
0432 

°433 

9-9570 
9569 
9569 
9568 

9567 

9-9567 
9566 
9566 

9565 
9564 

9.6286 
6289 
6292 
6295 
6297 

10.3714 

37" 

3708 

3705 
3703 

9.6720 
6723 
6726 
6729 
6733 

10.3280 
3277 
3274 
3271 
3267 

10.0433 
0434 
0434 
0435 
0436 

9.6300 
6303 
6305 
6308 
6311 

10.3700 

3697 
3695 
3692 
3689 

9.6736 
6739 
6743 
6746 
6749 

10.3264 
3261 
3257 
3254 
3251 

10.0436 
0437 
0437 
0438 

0439 

9.9564 
9563 
9563 
9562 
_  956  ' 
9-6561 
9560 
9560 
9559 
9558 

9-631  3 
6316 
6319 
6321 
6324 

10.3687 
3684 
3681 

3679 
3676 

9.6752 
6756 

6759 
6762 
6765 

10.3248 
3244 
3241 
3238 
3235 
10.3231 
3228 

3225 
3222 
3218 

10.0439 
0440 
0440 
0441 
0442 

9.6327 
6329 
6332 
6335 
6337 

10.3673 
3671 
3668 
3665 
3663 
10.3660 
3658 
3655 
3652 
_365o 

10.3647 

3644 
3642 

3639 
3636 

9.6769 
6772 

6775 
6778 
6782 

10.0442 
0443 
°443 
0444 

0445 

9-9558 
9557 
9557 
9556 
9555 

9.6340 
6342 

6345 
6348 

6350 

9.6785 
6788 
6791 

6795 
6798 

9.6801 

6811 
6814 

10.3215 
3212 

3209 
3205 
3202 

10.0445 
0446 
0446 

0447 
0448 

9-9555 
9554 
9554 
9553 
.    9552 

9-6353 
6356 
6358 
6361 
6364 

10.3199 
3196 
3192 

3189 
3186 

10.0448 
0449 
0449 
0450 

-JH5.L 
10.0451 
0452 
0452 
0453 
Q454 
10.0454 
0455 
°455 
0456 

Q457 
10.0457 
0458 
0458 

0459 
6460 

9-9552 
955i 
9551 
955° 
_?549 
9-9549 
9548 
9548 
9547 
9546 
9.9546 
9545 
9545 
9544 
9543 

9.6366 
6369 
6371 
6374 
6377 

10.3634 
3631 
3629 
3626 

3623 

9.6817 
6821 
6824 
6827 
6830 

10.3183 

3179 
3176 

3173 
3170 

9.6379 
6382 
6385 
6387 
6390 

10.3621 
3618 
3615 
3613 
3610 

10.3608 
3605 
3602 
3600 
3597 

9.6834 

6837 
6840 
6843 
6846 

10.3166 

3l63 
3160 

3157 
_3154 
10.3150 
3H7 
3J44 
3*4« 
3U7 

9.6392 

6395 
6398 
6400 
6403 

9.6850 

6853 
6856 
6859 
6863 

9-9543 
9542 
9542 
954i 
9540 

1$ 

6411 

6413 
6416 

9.6418 

io.3595 
3592 
3589 
35f7 
_35?4 
10.3582 

9.6866 
6869 
6872 
6875 
6879 
9.6882 

10.3134 
3131 

3128 

3125 
3121 

10.3118 

10.0460 
0461 
0462 
0462 
0463 
10.0463 

9.9540 
9539 
9538 
9538 
_953.7 
9-9537 

Siii. 

/ 

m    s 

Cos. 

O'.l     I8 

Diff. 

Sec. 

Cot. 

!• 

Tan. 

Cosec. 

m    s 

/ 

Diff. 

115°  =  7"  40'"  ]                                                                                                   [  4h  16m  =  64°  \ 

TABLE  IX. 


53 


26^ 

0 
1 

2 
3 
4 

5 

6 

7 

I 

!  10 
11 

12 
13 
14 
15 
16 
17 
18 
I  19 
20 
21 
22 
123 
i  24 
j  25 
!  26 
1  27 
i  28 
29 
30 
i  31 
!  32 
i  33 
34 
j  35 
36 
37 
38 
i  39 

|40 
11 
'42 
43 
11 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
J57 
|58 
!59 
60 

=  1°  44 

n,  j 

Sill. 

9.6418 
6421 
6424 
6426 
6429 
9.6431 
6434 
6437 
6439 
6442 

L,o&.  Sines,  Tangents,  and  Secants.       [  IO" 

12!tl  =  153° 

in    s 

44    O 
4 

8 
12 
16 
44  2O 
24 
28 
32 
36 
44  4O 
44 
48 
52 
56 
45    0 
4 
8 
12 
16 
45  2O 
24 
28 
32 
36 
4540 
44 
48 
52 
56 
46    O 
4 
8 
12 
16 
4620 
24 
28 
32 
36 
4640 
44 
48 
52 
56 
47    0 
4 
8 
12 
16 
4720 
24 
28 
32 
36 
47  4O 
44 
48 
52 
56 
48    0 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

10.0463 
0464 
0465 
0465 
0466 

Cos. 

9-9537 
9536 
9535 
9535 
__.9534 
9-9534 
9533 
9532 
9532 
9531 

ni     s 

60 
59 

58 
57 
56 
55  i 
54 
53 
52  i 
51  | 

50 
49 

48 
47  ! 
46 
45  i 
44  i 
43  i 
42  ! 
41 
40 
39 
38  i 
37  1 
36 
35 
34 
33  ! 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
2O 
19  ! 
18 
17  ! 
16  | 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

O'.l 

o 
i 

i 

2 
2 

2 
2 

O 

I 
I 
I 
I 
2 
2 
2 
2 

O 

T 

I 
I 

2 
2 
2 
2 

0 

I 

2 
2 
2 
2 

O 
I 
I 

I 

2 
2 
2 
2 

O 

I 

I 

2 
2 
2 
2 

O'.l 

I- 

2 
2 

I 
2 
2 

I 
2 
2 

I 

2 
2 

2 
2 

2 
2 

O'.l 

0 

i 
i 

1 

2 
2 

3 
3 
3 

o 
i 

i 
i 

2 
2 

3 
3 
3 

0 

i 
i 
i 

2 
2 

3 
3 
3 

o 
i 
i 
i 

2 

2 

3 
3 
3 

o 
i 

2 
2 

3 
3 
3 

0 

i 
i 
i 

2 
2 

3 
3 
3 

1s 

10.3582 
3579 
3576 
3574 
._..357JL 
10.3569 
35f 
3563 
356i 
3558 

9.6882 
6885 
6888 
6891 
6895 
9.6898 
6901 
6904 
6907 
6911 
"9.6914 

6917 
6920 

6923 
_6917 
9.6930 
6933 
6936 
6939 
6942 

i 

2 

'•J 

2 
2 

I 

2 
2 

2 
2 

I 
2 
2 

2 
2 

10.3118 

3H5 
3112 
3109 
3105 

16    O 
1556 
52 
48 
1544 
40 
36 
32 
28 
1524 
20 
16 
12 
8 
15    4 
15    0 
14  56 
52 
48 
1444 
40 
36 
32 
28 
1424 
2O 
16 
12 
8 
14    4 
14    0 
1356 
52 
48 
1344 
40 
36 
32 
28 
1324 
20 
16 
12 
8 
13    4 
13    0 
1256 
52 
48 
1244 
40 
36 
32 
28 
1224 
20 
16 
12 
8 
12    4 
12    0 

10.3102 

3°99 
3096 

3093 
3089 

10.0466 
0467 
0468 
0468 
0469 

9.6444 
6447 

6449 
6452 

_J>454_ 

9-6457 
6460 
6462 
6465 
6467 

!o-3556 

3553 
355J 
3548 
__3_546 

iQ-3543 
3540 
3538 
3535 
3533 

10.3086 
3083 
3080 

3°77 
3Q73 
10.3070 
3067 

3064 
3061 

3058 

10.0470 
0470 
0471 
0471 
_547?_ 
10.0473 

0473 
0474 

0475 
0475 

9-953° 
953° 
9529 
9529 
_9S?8 

9-9527 
9527 
9526 

9525 
9525 

9.6470 
6472 

6475 
6477 
6480 

10.3530 

3528 
3525 
3523 
3520 

9.6946 

6949 
6952 

6955 
6958 

10.3054 

3°5! 
3048 

3045 
3042 

10.3038 
3035 
3°32 
3029 
3026 

10.0476 
0476 
0477 
0478 
0478 

9.9524 
9524 
9523 
9522 
9522 

9.6483 
6485 
6488 
6490 
6493 

10.3517 
3515 
3512 
351° 
3507 

9.6962 
6965 
6968 
6971 
6974 

10.0479 
0480 
0480 
0481 
0481 

9.9521 
9520 
9520 
95*9 
9519 
9.95i8 
9517 
9517 
95i6 
9515 

9.6495 
6498 
6500 
6503 
6505 

10.3505 
3502 

35°° 
3497 
3495 

9.6977 
6981 
6984 

6987 
6990 

10.3023 
3019 
3016 

3013 
3010 

10.0482 
0483 
0483 
0484 
048-5 

9.6508 
6510 
6513 
65*5 
6518 

10.3492 
3490 
3487 
3485 
3482 

10.3479 
3477 
.  3474 
3472 
3469 

9-6993 
6996 
6999 
7003 
7006 

10.3007 

3°°4 
3001 
2997 
2994 

10.0485 
0486 
0487 
0487 
0488 

9-95  i  5 
95H 
9513 
9513 
95i2 

9.6521 

*$*l 
6526 

6528 
6531 

9.7009 
7012 

7015 
7018 
7022 

10.2991 

2988 
2985 
2982 
2978 

10.0488 
0489 
0490 
0490 
0491 

9.9512 

95" 
95io 
95io 
9509 

9-6533 
6536 
6538 
6541 
6543 

10.3467 

3464 
3462 

3459 
3457 

9.7025 
7028 
7031 
7034 
7037 

10.2975 
2972 
2969 
2966 
2963 

10.0492 
0492 

0493 
0494 
0494 

9.9508 
9508 

95°7 
9506 
9506 

9.6546 
6548 
655i 
6553 
6556 

10.3454 
3452 
3449 
3447 
3444 

9.7040 

7°43 
7047 
7050 
7053 

10.2960 
2957 
2953 
2950 
2947 

10.0495 

0495 
0496 
0497 
0497 

9.9505 
95°5 
95°4 
95°3 
95°3 

9-6558 
6561 

6563 
6566 
6568 

10.3442 
3439 
3437 
3434 
3432 

9.7056 

7059 
7062 
7065 
7069 

10.2944 
2941 
2938 
2935 
2931 

10.0498 
0499 
0499 
0500 
0501 

9.9502 
95°i 
95°i 
9500 

9499 

9.6570 

10.3430 

9.7072 

10.2928 

10.0501 

9-9499 

' 

j  

in    s 

Cos. 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

116°  =  7h  44m  ]                                                                                                    [  4h  12m  =  63° 

54 


TABLE  IX. 


27°=  lh  48:D]              Log.  Sines,  Tangents,  and  Secants.         [lOh  8m  =  152° 

i 

m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

Ill       S 

i 

O'.l 

1s 

O'.l 

1s 

i 

2 
2 

I 
2 
2 

I 

2 
2 

2 
2 

I 
2 
2 

I 
2 
2 

I8 

0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 

i  1~ 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
I  42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

48    0 
4 
8 
12 
16 
48  20 
24 
28 
32 
36 
4840 
44 
48 
52 
56 
49    0 

8 
12 
16 
49  2O 
24 
28 
32 
36 
49  4O 
44 
48 
52 
56 
50    0 
4' 
8 
12 
16 
5O2O 
24 
28, 
32 
36 
5040 
44 
48. 
52 
56 
51    0 
4 
8 
12 
16 
51  2O 
24 
28 
32 
36 
5140 
44 
48 
52 
56 
52    0 

9.6570 
6573 
J575 
6578 
6580 

0 

i 
i 
i 
i 

2 
2 
2 
2 

O 

1 
I 
I 
I 
2 
2 

2 
O 

I 
I 

2 

2 
2 

2 

0 
I 

I 
I 
I 
2 
2 
2 
2 

O 

I 
I 
I 
I 
2 
2 
2 
2 

0 
I 

I 

I 
2 
2 
2 
2 

i 

2 

2 

I 

2 

I 
I 
2 

2 

I 

I 
2 

10.3430 
34<7 
3425 
3422 
3420 

9.7072 

7075 
7078 
7081 
7084 

o 
i 

i 
i 

2 
2 

3 

3 
3 

o 

2 

2 

3 
3 
3 

o 
i 

i 
i 

2 
2 
3 

3 
3 

0 

I 
I 

2 
2 
3 

3 
3 

0 

2 
2 

'  3 
3 
3 

o 

2 

2 

3 
3 
3 

10.2928 

2925 
2922 
2919 
2916 

10.0501 
O5O2 
O5O2 
0503 
0504 
IO.O5O4 
0505 
0506 
0506 
0507 

9-9499 
9498 
9498 
9497 
9496 

12    0 
1156 
52 

48 
1144 
40 
36 
32 
28 
1124 
2O 
16 
12 
8 
11    4 
11    0 
1056 
52 
48 
1044 
40 
36 
32 
28 
1024 
20 
16 
12 
8 
1O    4 
10    0 
956 
52 
48 
944 
4O 
36 
32 
28 
924 
20 
16 
12 
8 
9    4 
9    O 
856 
52 
48 
844 
40 
36 
32 
28 
824 
20 
16 
12 
8 
8    4 
8    0 

00 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

T 

8 
7 
6 
5 
4 
3 
2  j 
1 

0 

9.6583 
6585 
6588 
6590 
6593 

10.3417 

3415 
3412 

34»o 
3407 

9.7087 
7090 

7093 
7097 
7100 

10.2913 
2910 
2907 

2903 
2900 

9.9496 
9495 
9494 
9494 
9493 
9.9492 
9492 
9491 
9490 
9490 

9-6595 

6598 
6600 
6603 
6605 

10.3405 

3402 
3400 
3397 
3395 

9.7103 
7106 
7109 
7112 
7"5 

10.2897 
2894 
2891 
2888 
2885 

IO.O5O8 
0508 
0509 
O5IO 
0510 

9.6607 
6610 
6612 
6615 
6617 

10-3393 
3390 
3388 
33f5 
^l8! 
10.3380 
3378 
3375 
3373 
337i 

9.7118 
7121 
7125 

7128 

7131 

10.2882 
2879 
2875 
2872 
2869 

IO.O5II 
O5I2 
0512 
0513 
0514 

9.9489 
9488 
9488 
9487 
9486 

9.6620 
6622 
6625 
6627 
6629 

9.6632 
6634 
6637 
6639 
6642 

9-7I34 
7137 
7140 

7143 
7146 

10.2866 
2863 
2860 
2857 
2854 

IO.O5I4 
0515 

°S1S 
0516 

0517 

9.9486 
9485 
9485 
9484 
9483 

10.3368 
3366 
3363 
336i 
3358 

9.7149 
7152 
7i56 
7i59 
7162 

10.281;  i 

2848 
2844 
2841 
2838 

io.o<U7 
0518 
0519 
0519 
0520 

9.9483 
9482 
9481 
9481 
9480 

9.6644 
6646 
6649 
6651 
6654 

10.3356 
3354 
335i 
3349 
3346 

9-7165 
7108 
7171 

7174 
7177 

10.2835 
2832 
2829 
2826 
2823 

10.0521 
0521 
0522 
0523 
0523 
10.0524 
0525 
0525 
0526 
0527 

9-9479 
9479 
9478 
'  9477 
9477 

9.6656 
6659 
6661 
6663 
6666 

10.3344 
3341 
3339 
3337 
3334 

9.7180 
7183 
7186 
7189 
7192 

10.2820 
2817 
2814 
2811 
2808 

9.9476 
9475 
9475 
9474 
9473 

9.6668 
6671 
6673 
6675 
6678 

10.3332 
3329 
3327 
3325 
3322 

9.7196 
7199 
7202 
7205 
7208 

10.2804 
2801 
2798 

'2795 
2792 

10.0527 
0528 
0529 
0529 
_°13?. 
10.0531 

053  I 
0532 
0533 
0533 

9-9473 
9472 

947i 
947i 
9470 

9.6680 
6683 
6685 
6687 
6690 

10.3320 
3317 
3315 
33*3 
33io 

9.7211 
7214 

7217 
7220 
7223 

10.2789 
2786 

2783 
2780 
2777 

9.9460 
9469 
9468 
9467 

9467 

9.6692 

g95 
6697 

6699 
6702 

^6704 
6707 
6709 
6711 
6714 

10.3308 
3305 
33°3 
33oi 
3298 
10.3296 

3293 
3291 
3289 
3286 

9.7226 

7229 
7232 

7235 
7238 

10.2774 

2771 
2768 
2765 
2762 

10.0534 
0535 
°535 
0536 

°537 
10.0537 
0538 
0539 
0539 
0540 

9.9466 
9465 
9465 
9464 
9463 

9.7241 

7245 
7248 

7251 
7254 

10.2759 

2755 
2752 

2749 
2746 

9-946,3 
9462 

946i 
9461 

9460 

9.6716 

10.3284 

9-7257 

10.2743 

10.0541 

9-9459 

' 

m    s 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l 

Tan. 

Cosec. 

Sin. 

m     a 

/ 

Diff. 

Diff. 

117°  =  7h48ra]                                                                                                       [4h8m  =  02 

TABLE  IX. 


55 


28D  -  lh  52tn  ]            I^og.  Sines,  Tangents,  and  Secants.        [  lOb  4m  =  151° 

' 

m     s 

52    O 
4 
8 
12 
16 
52  2O 
24 
28 
32 
36 
5240 
44 
4§ 
52 
56 
53    O 
4 
8 
12 
16 
53  20 
24 
28 
32 
36 
534O 
44 
48 
52 
56 
54    0 
4 
8 
12 
16 
54  2O 
24 
28 
32 
36 
54  4O 
44 
48 
52 
56 
55    0 
4 
8 
12 
16 
5  5  2O 
24 
28 
32 
36 
55  40 
44 
48 
52 
56 
56    0 

Sin. 

9.6716 
6718 
6721 
6723 
6726 

Di 
O'.l 

ff. 

~i"~ 

Cosec. 

Tan. 

9-7257 
7260 
7263 
7266 
7269 

Di 
O'l 

1s 

i 

2 
2 

2 
2 

2 
2 

2 
2 

I 

2 
2 

I 

2 
2 

Cot. 

10.2743 
2740 
2737 
2734 
2731 

Sec. 

10.0541 

0541 

0542 

0543 
054^ 

Cos. 

m    s 

' 

0 

1 
2 
3 
4 
5 
6 
7 
8 

•  9 

i  10 

I  *  8 
1  12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
24 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

o 

: 

i 
i 

2 

2 
2 
2 

0 
I 

I 
I 
I 

2 
2 
2 

2 

O 

I 

2 
2 
2 

2 

O 
I 

I 
I 
I 

2 
2 

2 
2 

O 
I 
I 

I 

2 
2 
2 

2 

O 
I 

2 
2 
2 
2 

i 

\ 

i 
i 

•2 

I 
I 

2 

2. 

I 

I 
2. 

2 

10.3284 
3282 
3279 
3277 
3274 

o 
i 

i 

i 

2 
2 

3 
3 
3 

o 

i 
i 

i 

2 

2 

3 
3 
3 

o 
i 

i 

2 
2 

3 
3 
3 

o 
i 
i 

2 
2 

3 
3 
3 

o 

2 
2 
3 

3 
3 

o 
I 

I 
I 

2 

2 

3 
3 
3 

9-9459 
9459 
945s 
9457 
9457 

8    0 
756 
52 
48 
744 
40* 
36 
32 
28 
724 
20 
16 
12 
8 
7    4 
7    0 
656 
52 
48 
644 
4O 
36 
32 
28 
624 
20 
16 
12 
8 
6    4 
6    O 
556 
52 
48 
544 
40 
36 
32 
28 
5*24 
20 
16 
12 
8 
5    4 
5    0 
456 
52 
48 
444 
40 
36 
32 
28 
424 
20 
16 
12 
8 
4    4 
4    0 

60 
59 
58 
57 
56 
55 
54 
53  i 
52 
51  i 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 

2O 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8 
7 
6 
5 
4 
3 
2 
1 

O 

9.6728 
6730 
6733 
6/35 
6737 

10.3272 
3270 
3267 
3265 
3263 

9.7272 

7275 
7278 
7281 
7284 

10.2728 
2725 
2722 
2719 
2716 

10.0544 
0545 
0545 
0546 

0547 

9.9456 
9455 
9455 
9454 
9453 

9.6740 
6742 
6744 
6747 
6749 
9.6752 

& 

6759 
6761 

10.3260 
3258 
3256 
3253 
325  i 
10.3248 
3246 
3244 
3241 
3239 

9.7287 
7290 

7293 
7296 

7299 
9.7302 

7305 
7308 

73" 
73  H 

10.2713 
2710 
2707 
2704 
2701 
10.2698 
2695 
2692 
2689 
2686 

10.0547 
0548 
0549 
0549 
055° 
10.0551 
055  i 
0552 
0553 
0553 

9-9453 
9452 
945  i 
945  i 
945° 
9-9449 
9449 
9448 

9447 
9447 

9.6763 
6766 
6768 
6770 
6773 

10.3237 
3234 
3232 
323° 
3227 

9-73I7 

7320 

7324 
7327 
7330 

10.2683 
2680 
2676 
2673 
2670 

10.0554 
0555 
0556 
0556 
0557 

9.9446 
9445 
9444 
9444 
9443 

9-6775 
6777 
6780 
6782 
6784 

10.3225 

3223 
3220 
3218 
3216 

9-7333 
7336 
7339 
7342 

7345 

10.2667 
2664 
2661 
2658 
2655 

10.0558 
0558 
0559 
0560 
0560 

9.9442 
9442 
9441 

944° 
9440 

9.6787 
6789 
6791 

6794 
6796 

10.3213 
3211 
3209 
3206 
3204 

97348 
735  i 
7354 
7357 
7360 

10.2652 
2649 
2646 
2643 
2640 

10.0561 
0562 
0562 

0563 
0564 

9-9439 
9438 
9438 
9437 
9436 

9.6798 
6801 
6803 
6805 
6808 

10.3202 
3*99 
3r97 
3195 
3192 

9-7363 
7366 

7369 

7372 
7375 

10.2637 
2634 
2631 
2628 
2625 

10.0564 
0565 
0566 
0567 
0567 

9.9436 
9435 
9434 
9433 
9433 
9-9432 
943i 
943  i 
943° 
9429 

9.6810 
6812 
6814 
6817 
6819 

10.3190 
3188 
.    3186 
3183 
3181 

9-7378 
738i 
7384 
7387 
7390 

10.2622 
2619 
2616 
2613 
2610 

10.0568 
0569 
0569 
0570 
O57i 

9.6821 
6824 
6826 
6828 
6831 

10.3179 
3176 
3*74 
3172 
3169 

9-7393 
7396 
7399 
7402 

7405 

10.2607 
2604 
2601 
2598 
2595 
10.2592 
2589 
2586 

2583 
2580 

10.0571 
0572 
°573 
0573 
0574 

9.9429 
9428 
9427 
9427 
9426 

9-6833 
6835 
6837 
6840 
6842 

10.3167 
3^5 
3163 
3160 
3158 

9.7408 
7411 
74H 

7417 
7420 

10.0575 
0576 
0576 

0577 
0578 

9.9425 
9424 
9424 
9423 
9422 

9.6844 
6847 
6849 
6851 
6853 

10.3156 
3153 
3i5i 
3H9 
3H7 

9-7423 
7426 
7429 
7432 
7435 

10.2577 

2574 
257i 
2568 

2565 

10.0578 

0579 
0580 
0580 
0581 

9.9422 
9421 
9420 
9420 
9419 
9.9418 

9.6856 

10.3144 

9-7438 

10.2562 

10.0582 

i 

m    s 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

118°  =  7h  52m  ]                                                                                                       [  4h  4ra  ==  61° 

56 


TABLE  IX. 


29 

/ 

fc  lh  56 
m    s 

11  ] 
Sin. 

Log.  Sines,  1 

Diff. 
Cosec. 
0.1     1 

range 

T.m. 

«i«».  and  Scca 

Diff. 
—    Cot. 
O'.l     1s 

LlltS. 

Sec. 

[10 

Cos. 

1  Om  =  150° 

m    s 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
11 
15 
16 
17 
18 
19 
20 
21 
22 
23 
21 
25 
26 
27 
!  28 
29 
30 
31 
32 
33 
31 
35 
36 
37 

:  38 

39 
40 
11 
12 
i  43 
44 
45 
46 
;  47 
48 
49 
50 
51 
52 
53 
51 
55 
56 
57 
58 
59 
60 

56    0 
4 

8 
12 
16 
56  20 
21 
28 
32 
36 
5640 
11 
48 
52 
56 
57    0 
4 
8 
12 
16 
5  7  2O 
24 
28 
32 
36 
5740 
44 
48 
52 
56 
58    0 
4 
8 
12 
16 
5820 
24 
28 
.32 
36 
5840 
44 
48 
52 
56 
59    O 
4 
8 
12 
16 
59  2O 
24 
28 
32 
36 
5940 
44 
48 
52 
56 
60    0 

9.6856 
0858! 
6800  1 
6863  i 
6865 

9.6867  ; 
6869 

6872  i 
6874 
63  76 

0          1 

i        j 

1          2 
I 

2  ; 

2 

C)          1 
1           I 

I          2 
I    ' 
I    j 

2  ; 

2 

o  !     i 

2 

2    j 

S 

d  : 

r> 
| 

j 

2 
2.    j 

O           I 

1           ' 
I    j      2. 

i  i 
i 

I 

2 
2 
2 

O           I 
I           I 

I          2 
I    ! 
I 
I 

2 

2 

10.3144 
3H2 
3MO 
3137 

Mi 

10.3133 

& 

3126 
3124 

9-7438 
7440 

7443 
7446 

7449 

9-7452 
7455 
7458 
7461 
7464 

o 

i 
i 

2 
2 
2 

3 

o 

i 

2 
2 
2 

3 

0 

i 
i 
i 
i 

2 
2 
2 

3 
o 

I 

I 

2 

2 

•       2 

3 

0 

i 
i 

2 

2 
2 

3 

0 

2 
2 
2 

3 

2 
2 

2 
2 

2 

2 

I 
2 
2 

I 
2 
2 

i 

2 

2 

10.2562 
2560 
2557 
2554 

2551 
10.2548 

2545 
2542 
2539 
2536 

10.0582 
0583 

0583 
0584 
0585 
10.0585 
0586 
0587 
0587 
0588 

9.9418 
9417 
9417 
9416 

9415 

9-94I5 
9414 

9413 
9413 
9412 

4    0 

356 
52 

'48 
314 

4O 
36 
32 
28 
324 
20 
16 
12 
8 
3    4 
3    0 
256 
52 
48 
244 
40 
36 
32 
28 
224 
20 
16 
12 
8 
2    4 
2    0 
156 
52 
48 
144 
40 
36 
32 
28 
124 
20 
16 
12 
8 
1     4 
1    0 
056 
52 
48 
044 
4O 
36 
32 
28 
024 
20 
16 
12 
8 
0    4 
O    O 

6O 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
4O 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

9.6878 
6  38  1 
638} 
6885 
633? 
9.6890 
6892 

6894 
6896 
6899 
9.6901 
6903 
6905 
6908 
6910 

9.6912" 
6914 
6917 
6919 
6921 

10.3122 

3"9 
3"7 

3H5 
3  "3 
10.3110 
3108 
3106 

3I04 
3101 

10.3099 
3°97 
3°95 
3092 
3090 

9-7467 
7470 

7473 
7476 
_7479_ 
9.7482 

7485 
7488 

749i 
__749£ 

9-7497 
7500 

7503 
7506 

7509 

10.2533 
2530 
2527 
2524 

._?5*L 

10.2518 

2515 
2512 
2509 
2506 
10.2503 
2500 

2497 
2494 
2491 

10.0589 
0590 
0590 
0591 

__°!?2_ 

10.0592 

0593 
0594 
0594 
0595 

10.0596 

0597 
0597 
0598 

0599 

9.9411 
9410 
9410 
9409 
9408 
9.9408 

9407 
9406 
9406 
9405 
9.9404 
9403 
9403 
9402 
9401 

10.3088 
3086 

3083 
3081 

3°79 

9-7512 

7515 
75i8 
752i 
7523 

10.2488 

2485 
2482 

2479 
2477 

10.0599 

0600 
0601 
0602 
0602 

9.9401 
9400 
9399 
939f 
9398 

9.6923 
6926 
6928 
6930 
6932 
9-6935 
6937 
6939 
6941 

6943 

10.3077 

3°74 
3072 
3070 
3068 

9.7526 
7529 
7532 
7535 
7538 

10.2474 
2471 
2468 

2465 
2462 

10.0603 
0604 
0604 
0605 
0606 

9-9397 
9396 
9396 
9395 
_9394 

9-9393 
9393 
9392 
9391 

9-9390 
9389 

92» 

9388 

9387 

10.3065 

3063 
3061 

3059 
3057 

9-7541 
7544 
7547 
755° 
7553 

10.2459 
2456 
2453 
2450 
2447 

10.0607 
0607 

0608 

0609 
0609 

9.6946 
6948 
6950 
6952 
691L 
9.6957 

6959 
6961 
6963 
6966 
9.6968" 
6970 
6972 
6974 

6977. 
9.6979 
6981 

6983 
6985 
6988 
9.6990 

10.3054 
3052 
3050 
3048 

._3?45_ 
10.3043 
3°4i 
3039 
3°37 
3°34 

9-7556 
7559 
7562 

£3 

10.2444 
2441 

2438 
2435 
2432 

1  0.06  10 

0611 

0612 
0612 
0613 

9-7571 

7573 
7576 
7579 
7582 

10.2429 
2427 
2424 
2421 
2418 

10.0614 
0615 
0615 
0616 
0617 

9.9386 
9385 
93f5 
9384 
9383 

10.3032 

3030 
3028 
3026 
3023 
10.3021 
3019 
3017 

3OI5 
3012 

10.3010 

9-7585 
7588 
7591 
7594 
7597 

10.2415 
2412 
2409 
2406 
2403 

10.0617 
0618 
0619 
0620 
0620 

9.9383 
9382 

938i 
9380 
9380 

9.  7600 
7603 
7606 
7609 
7611 
"97614" 

10.2400 
2397 
2394 
2391 
2389 
10.2386 

10.0621 
0622 
0623 
0623 
0624 
10.0625 

9-9379 
9378 
9377 
9377 
9376 

9-9375 

/ 
i 

m    B 

Cos. 

O'.l     1-' 

Sec. 

Cot. 

O'.l 

1s 

5T 

Tan. 

Cosec. 

Sin.    1   m     s 

/ 

Diff. 

Di 

119^--7h  56"'  J 

[4h  0"\=60° 

TABLE  IX. 


57 


'  30° 

t:  2h  Om  ]              L-og.  Sines,  Tangents,  and  Secants.         [  9h  56'"  =  149° 

'        m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.      Cos. 

m     s 

60    0 
5956 
52 
48 
59  44 
40 
36 
32 
28 
5924 
20 
16 
12 
8 
59    4 
59    O 
5856 
52 
48 
5844 
40 
36 
32 
28 
58  24 
20 
16 
12 
8 
58    4 
58    0 
5756 
52 
48 
5744 
40 
36 
32 
28 
5724 
20 
16 
12 
8 
57    4 
57    0 
5656 
52 
48 
5644 
40 
36 
32 
28 
5624 
20 
16 
12 
8 
56    4 
56    0 

i 

0.1     1s 

OM 

1s 

O 

1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
j  35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
i  47 
148 
49 
50 
51 
52 
53 
54 

;  55 

i  56 
57 
58 
59 
6O 

0    0 
4 

8 
12 
16 
020 
24 
28 
32 
36 
O4O 
44 
48 
52 
56 
1     0 
4 
8 
12 
-    16 
12O 
24 
28 
32 
36 
140 
44 
48 
52 
56 
2    O 
4 
8 
12 
16 
2  2O 
24 
28 
32 
36 
24O 
44 
48 
52 
56 
3    0 
4 
8 
12 
16 
3  2O 
24 
28 
32 
36 
340 
44 
48 
52 
56 
4    0 

j 
m    s 

! 

9.6990 
6992 
6994 
6996 
6998 

o 
o 

2 
2 

O 
O 

2 

2 

0 
O 

2 

2 

0 
O 

2 

2 

O 
O 

2 
2 

i 

i 

2 

I 
I 

2 

I 
2 

2 

2 

I 

I 
2 

10.3010 
3008 
3006 
3004 
3002 

9.7614 
7617 
7620 
7623 
7626 

o 
J 

2 

2 

3 

o 
I 
I 
I 
I 

2 
2 
2 

3 
o 

I  ' 

2 

2 
2 

3 

0 

I 

I 

I 

2 
2 
2 
3 

0 

I 

2 
2 
2 

3 

o 
i 

i 
i 

2 
2 
2 

3 

i 

2 
2 

I 
2 
2 

I 

2 
2 

I 

2 
2 

I 
2 
2 

I 
2 
2 

10.2386 

2383 
2380 

2377 
2374 

10.06215 
0625 
0626 
0627 
0628 

9-9375 
9375 
9374 
9373 
9372 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

9.7001 
7003 
7005 
7007 
7009 

10.2999 
2997 
2995 
2993 
2991 

9.7629 
7632 

7635 
7638 
7641 

10.2371 
2368 

2365 
2362 

2359 
10.2356 
2354 
235J 

2348 

2345 

10.0628 
0629 
0630 
0631 
0631 

9-9372 
9371 
937" 
9369 
9369 

9.7012 
7014 
7016 
7018 
7020 

10.2988 
2986 
2984 
2982 
2980 

9.7644 
7646 
7649 
7652 
7655 

10.0632 

0633 
0033 
0634 
°635 

9.9368 
9367 
9367 
9366 

9365 

9.7022 
7025 
7027 
7029 
7031 

10.2978 
2975 
2973 
2971 
2969 

9.7658 
7661 
7664 
7667 
7670 

10.2342 
2339 
2336 
2333 
233° 

10.0636 
0636 
0637 
0638 
0639 

9.9364 
9364 
9363 
9362 
9361 

9-7033 
7035 
7037 
7040 

7042 

9.7044 
7046 
7048 
7050 
7°5JL 
9-7°55 
7057 
7059 
7061 
7063 

10.2967 
2965 
2963 
2960 
2958 

9-7673 
7675 
7678 
7681 
7684 

10.2327 

2325 
2322 

2319 
2316 

10.0639 
0640 
0641 
0642 
0642 

9.9361 
9360 
9359 
9358 
9358 

10.2956 

2954 
2952 

295° 
2947 

9.7687 
7690 

7693 
7696 
7699 

10.2313 
2310 
2307 
2304 
2301 

10.0643 
0644 
0645 
0645 
0646 

9-9357 
9356 
9355 
9355 
9354 

10.2945 

2943 
2941 

2939 
2937 

9.7701 

7704 
7707 
7710 
7713 

10.2299 
2296 
2293 
2290 
2287 

10.0647 
0648 
0648 
0649 
0650 

9-9353 
9352 
9352 

935  i 
935° 

9.7065 
7068 
7070 
7072 
7074 

10.2935 
2932 
2930 
2928 
2926 

9.7716 

7719 

7722 

7725 
7727 

10.2284 
2281 
2278 
2275 
^73 
10.2270 
2267 
2264 
2261 
2258 

10.0651 
0651 
0652 

0653 
0654 

9-9349 
9349 
9348 
9347 
9346 

9.7076 
7078 
7080 
7082 
7085 

10.2924 
2922 
2920 
2918 
2915 

9-773° 
7733 
7736 
7739 
7742 

10.0654 

0655 
0656 
0657 
0657 

9-9346 
9345 
9344 
9343 
9343 

9.7087 
7089 
7091 
7093 
7095 

10.2913 
2911 
2909 
2907 
2905 

9-7745 
7748 
775° 
7753 
7756 

10.2255 
2252 
2250 
2247 
2244 

10.0658 

°659 
0660 
0660 
0661 

9.9342 
934i 
9340 
9340 
9339 

9.7097  ! 
7099        o 
7102  1      o 

7104      I 

7106        I 

10.2903 
2901 
2898 
2896 
2894 

9-7759 
7762 

7765 
7768 
7771 

10.2241 
2238 

2235 
2232 
2229 

10.0662 
0663 
0663 
0664 
0665 

9-9338 
9337 
9337 
9336 
9335 

9.7108 
7110 

7112 
7114 

7116 

I 

I 
I 

2 

10.2892 
2890 
2888 
2886 
2884 

9-7773 
7776 

7779 
7782 

7785 

10.2227 
2224 

2221 
22l8 

22*5 

IO.22I2 

10.0666 
0666 
0667 
0668 
0669 

9-9334 
9334 
9333 
9332 
9331 

9.7118 

10.2882 

9.7788 

10.0669 

9-9331 

|     , 

Cos. 

0.1 

1s 

Sec. 

1 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

;  12O°  M  8h  O'»  ]                                                                                                       [  3h  56m  =  59° 

58 


TABLE  IX 


31°      \>     I                   L,og.  Sines,  Tangents,  and  Secants.         [  9h  52m  =  148° 

/ 

O 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

m    & 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

Ill      S 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48 
47 
46 
45 
44 
43 
42 
41 

40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28  : 
27 
26 
25 
24  ! 
2J{ 
22 
21 
20 
19 
18 
17 
16 
15 
14  i 
13 
12 
11 
10 
9 
8 
7 
i 

•I 

3 
2 

1 

o 

i 

O'.l 

I8 

i 
i 

2 

I 

1 

2 

I 

2 

I 

2 

2 

I 
I 

2 

O'.l     I8 

4    0 
4 

8 
12 
16 
420 
24 
28 
32 
36 
440 
44 
48 
52 
56 
5    0 
4 
8 
12 
16 
520 
24 
28 
32 
36 
540 
44 
48 
52 
56 
6    0 
4 
8 
12 
16 
620 
24 
28 
32 
36 
640 
44 
48 
52 
56 
7    0 
4 
8 
12 
16 
7  2O 
24 
28 
32 
36 
74O 
44 
48 
52 
56 
8    0 

9.7118 
7120 
7123 
7125 
7127 

o 
o 

I 
I 
I 

I 
I 

2 
2 

0 
O 

2 

2 

O 
O 

I 
I 
I 
I 
I 
2 
2 

O 
0 
1 

I 
I 
I 
I 
2 
2 

0 
O 

2 
2 

0 
O 
1 
,  I 
I 
I 
I 
2 
2 

10.2882 
2880 
2877 
2875 
2873 

9.7788 
7791 

7793 
7796 

7799 

o 

i 
i 
i 
i 

2 
2 
2 
3 

0 

I 

2 
2 
2 
3 

0 

I 

I 

2 

2 
2 

3 

o 

i 
i 
i 
i 

2 
2 
2 

3 

0 

I 

2 

2 

2 

3 
o 

2 

2 
2 

3 

i 

2 

2 

I 

2 

2 

2 
2 

2 

2 

I 
2 
2 

2 
2 

IO.22I2 
22O9 
2207 
22O4 
22OI 

10.0669 
0670 
0671 
0672 
0672 

9-9331 
9330 
9329 
9328 
9328 

56    0 
55  56 
52 
48 
5544 
40 
36 
32 
28 
5524 
20 
16 
12 
8 
55    4 
55    0 
5456 
52 
48 
5444 
40 
36 
32 
28 
54  24 
20 
16 
12 
8 
54    4 
54    0 
53  56 
52 
48 
5341 
40 
36 
32 
28 
53  24 
20 
16 
12 
8 
53    4 
53    0 
52  56 
52 
48 
5214 
40 
36 
32 
28 
5224 
20 
16 
12 
8 
52    4 
52    0 

9.7129 
7I31 
7U3 
7135 
7137 

10.2871 
2869 
2867 
2865 
2863 
10.2861 
2859 
2856 
2854 
2852 
10.2850 
2848 
2846 
2844 
2842 

9.7802 

7805 
7808 
7811 
7813 

10.2198 

2195 
2192 
2189 
2187 

10.0673 
0674 
0675 
0675 
0676 

9-9327 
9326 

9325 
9325 
9324 

9-7I39 
7141 

7H4 
7146 
7148 

9.7816 

7819 
7822 
7825 
7828 

IO.2I84 

2181 

2178 

2175 
2172 

10.0677 
0678 
0678 
0679 
0680 

9-9323 
9322 
9522 
9321 
9320 

9-7I50 
7152 
7154 
7156 
7158 

9-7831 
7833 
7f36 
7839 
7842 

10.2169 
2167 
2164 
2161 

2158 

1  0.068  1 
0682 
0682 
0683 
0684 

9W9 
93i8 
93^8 
9317 
93i6 

9.7160 
7162 
7164 
7166 
7168 

10.2840 

2838 
2836 

2834 
2832 

9-7845 

7848 

7850 
7853 
785<> 

10.2155 

2152 
2150 
2147 
2144 

10.0685 
0685 
0686 
0687 
0688 

9-93I5 
9315 
93H 
9313 
9312 

9.7171 
7173 
7175 
7177 
7179 
9.7181 
7183 
7185 
7187 

_2!?9 

9.7191 
7193 
7195 
7197 

7199 
9.7201 
7203 
7205 
7208 
7210 

10.2829 
2827 
2825 
2823 
2821 

9-7859 
7862 
7865 
7868 
7870 

10.2141 

2138 

2135 
2132 

2130 

10.0688 
0689 
0690 
-   0691 
0692 

9.9312 

93" 
9310 

93°9 
_^3o8 

9.9308 

93°  7 
9306 

93°5 
93°5 

10.2819 
2817 
2815 
2813 
2811 

9-7873 
7876 
7879 
7882 
7885 

10.2127 
2124 

2121 

2118 

2115 

10.0692 
0693 
0694 
0695 
0695 

10.2809 
2807 
2805 
2803 
2801 
10.2799 
2797 
2795 
2792 
2790 

9.7887 
7890 

7893 
7896 
7899 

10.2113 
2IIO 
2107 
2104 
2IOI 

10.0696 
0697 
0698 
0699 
0699 

9.9304 

93°3 
9302 
9301 
9301 

9.7902 
7904 
7907 
7910 
7913 

10.2098 
2096 

2093 
2O9O 
2087 

10.0700 
0701 
0702 
0702 
0703 

9.9300 

9299 
9298 
9298 
9297 

9.7212 
7214 
7216 
7218 
7220 

9.7222 
7224 
7226 
7228 
7230 

10.2788 
2786 
2784 
2782 
2780 

9.7916 
7918 
7921 
7924 
7927 

IO.2O84 
2082 
2079 
2076 
2073 

10.0704 
0705 
0706 
0706 
0707 

9.9296 
9295 
9294 
9294 
9293 

10.2778 
2776 
2774 
2772 
2770 

9-7930 
7933 
7935 
7938 
794i 

1O.2O7O 
2067 
2065 
2062 
2059 

10.0708 
0709 
0709 
0710 
0711 

9.9292 
9291 
9291 
9290 
9289 

9.7232 

7236 

7238 
7240 

10.2768 
2766 
2764 
2762 
2760 

9.7944 
7947 
7949 
7952 
7955 

10.2056 

2053 
2051 
2048 
2045 

10.0712 
0713 

07i3 
0714 

0715 

9.9288 
9287 
9287 
9286 
9285 

9.7242 

10.2758 

9.7958 

IO.2O42 

10.0716 

9.9284 

/ 

m    & 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l 

1" 

Tan. 

Cosec. 

Sin. 

m    s 

Diff. 

Diff. 

i  121°  =  8h  4m  ]                                                                                                   .    [  3h  52'"  =  58° 

TABLE  IX. 


32°  =  2a  8m  ]              L,og.  Sines,  Tangents,  and  Secants.         [9h  481U  =  1473 

' 

m    & 

Sin. 

Diflf. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m     s 

' 

O'.l 

1s 

O'.l 

o 
i 

i 
i 
i 

2 
2 
2 
3 

0 

I 
I 
I 
I 

2 
2 
2 
3 

O 

I 
I 
I 

I 

2 
2 
2 

3 

o 
I 
I 
I 
I 

2 
2 
2 

3 
o 

2 

2 
2 

•    3 

0 

I 
I 
I 
I 

2 

2 
2 
3 

1s 

0 
1 

2 
3 
4 

5 
6 

7 

8 

» 
10 

I  11 
!  12 
!  13 
14 
15 
I  16 
17 
18 
19 
20 
21 
22 
1  23 
24 
25 
26 
27 
i  28 
29 
30 

i  31 
32 
33 
34 

35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
,48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

8    0 
4 
8 
12 
16 
820 
24 
28 
32 
36 
840 
44 
48 
52 
56 
9    0 
4 
8 
12 
16 
920 
24 
28 
32 
36 
94O 
44 
48 
52 
56 
10    0 
4 
8 
12 
16 
1O2O 
24 
28 
32 
36 
1040 
44 
48 
52 
56 
11    0 
4 
8 
12 
16 
1120 
24 
28 
32 
36 
1140 
44 
48 
52 
56 
12    0 

9.7242 
7244 
7246 
7248 
7250 

o 
o 

2 
2 

O 
O 

2 
2 

O 
O 

2 

0 
O 

2 
2 

0 
O 

2 
2 

0 
O 
I 
I 
I 
I 
I 
2 
2 

i 

2 

I 

I 
2 

I 
I 

2 

I 
I 

2 

2 

I 

I 
2 

~1*~ 

10.2758 
2756 

2754 
2752 
2750 

9-7958 
7961 

7964 
7966 
7969 

2 
2 

I 
2 
2 

2 
2 

2 
2 

.  I 

2 
2 

I 
2 
2 

10.2042 
2039 
2036 
2034 
2031 

10.0716 
0717 
0717 
0718 
0719 

9.9284 
9283 
9283 
9282 
9281 

52    0 
5156 
52 

48 
5144 
40 
36 
32 
28 
5124 
20 
16 
12 
8 
51    4 
51    0 
5056 
52 
48 
5044 
40 
36 
32 
28 
5024 
20 
16 
12 
8 
5O    4 
50    0 
4956 
52 
48 
4944 
40 
36 
32 
28 
4924 
20 
16 
12 
8 
49    4 
49    0 
4856 
52 
48 
4844 
4O 
36 
32 
28 
4824 
20 
16 
12 
8 
48    4 
48    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
5O 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18  i 
17 
16 
15 
14 
13 
12 
11 

10 

.9 

8  i 

I 

4 
3 
2 
1 

°i 

9.7252 

7254 
7256 

7258 
7260 

10.2748 
2746 

2744 
2742 
2740 

9.7972 

7975 
7978 
7980 

7983 

10.2028 
2025 

2022 
2O2O 
2017 

10.0720 
0721 
0721 
0722 
0723 

9.9280 

9279 
9279 

9278 
9277 

9.7262 
7264 
7266 
7268 
7270 

10.2738 
2736 

2734 
2732 
2730 

9.7986 
7989 
7992 
7994 
7997 

IO.2OI4 
201  1 

2008 
2OO6 
2003 

10.0724 
0725 
0725 
0726 
0727 

9.9276 
9275 
9275 
9274 

9273 

9.7272 

7274 
7276 
7278 
7280 

10.2728 
2726 
2724 
2722 
2720 

9.8000 
8003 
8006 
8008 
8011 

IO.2OOO 
1997 
1994 
1992 
1989 

10.0728 
0728 
0729 
0730 
0731 

9.9272 
9272 
9271 
9270 
9269 

9.7282 
7284 
7286 
7288 
7290 

10.2718 
2716 
2714 
2712 

2710 

9.8014 
8017 
8020 
8022 
8025 

10.1986 

1983 
1980 
I978 
1975 

10.0732 
0732 
°733 
0734 
°735 

9.9268 
9268 
9267 
9266 
9265 

9.7292 
7294 
7296 
7298 
7300 

10.2708 
2706 
2704 
2702 
2700 

9.8028 
8031 
8034 
8036 
8039 

10.1972 
1969 
1966 
1964 
1961 

10.0736 
0736 

0737 
0738 

0739 

9.9264 
9264 
9263 
9262 
9261 

9.7302 

7304 
7306 
73o8 
7310 

10.2698 
2696 
2694 
2692 
2690 

9.8042 
8045 
8047 
8050 
8053 

10.1958 

*955 
1953 
195° 
1947 

10.0740 
0741 
0741 
0742 
0743 

9.9260 
9259 
9259 
9258 

9257 

9.7312 
73H 
73  16 
73i8 
7320 

10.2688 
2686 
2684 
2682 
2680 

9.8056 

8059 
8obi 
8064 
8067 

10.1944 
1941 

!939 
1936 

J933 

10.0744 
0745 
0745 
0746 
0747 

9.9256 
9255 
9255 
9254 
9253 

9.7322 

7324 
7326 

7328 
7330 

10.2678 
2676 
2674 
2672 
2670 

9.8070 
8072 

8075 
8078 
8081 

10.1930 
1928 

1925 
1922 
1919 

10.0748 
0749 
0749 
0750 
0751 

9.9252 
9251 
9251 
9250 
9249 

9-7332 
7334 
733^ 
7338 
7340 

10.2668 
2666 
2664 
2662 
2660 

9.8084 
8086 
8089 
8092 
8095 

10.1916 
1914 
1911 
1908 
1905 

10.0752 
°753 
0753 
0754 
0755 

9.9248 
9247 

9247 
9246 

9245 

9-7342 
7344 
7345 
7347 
.7349 

10.2658 
2656 
2655 

265i 
10.2649 

2647 
2645 
2643 

2641 

9.8097 
8100 
8103 
8106 
8109 

10.1903 
1900 
1897 
1894 
1891 

10.0756 
0757 
0758 
0758 
0759 

9.9244 

9243 
9242 
9242 
9241 

9-7351 
7353 
7355 
7357 
7359 

9.8111 
8114 
8117 
8120 
8122 

10.1889 
1886 
1883 
1880 
1878 

10.0760 
0761 
0762 
0762 
0763 

9.9240 

9239 
9238 
9238 
9237 

9-736i 

10.2639 

9.8125 

10.1875 

10.0764 

9.6236 

.  / 

m    a 

Cos. 

0.1 

Sec. 

Cot. 

OM 

la 

Tan. 

Cosec. 

Sin. 

111      S 

/ 

Diff. 

Diff. 

122°  =  8h  8m  J                                                                                                       [  3h  48m  =  57^  ; 

60 


TABLE  IX. 


33°  ^2U  12s"  ] 

Log:.  Sines,  Tangents,  and  Secants.         [9b   S  S    =  146 

i    m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.      Cos. 

m     s 

' 

O'.l     1 

O'.l 

o 
i 

i 

i 
i 

2 
2 
2 

3 
o 

2 
2 
2 

3 

0 

i 
i 
i 

2 
2 
2 
3 

O 

2 
2 
2 

3 
o 

i 

i 
i 

2 

2 

2 

3 

0 

I 
I 

I 

2 
2 
2 

3 

1* 

0 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
18 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 

:  28 

29 
30 
31 
32 
33 
34 
35 
i  36 
I  37 

;  38 

!  39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
156 
57 
58 
59 
60 

12    0 
4 

8 
12 
16 
1220 
24 
28 
32 
36 
1240* 
44 
48 
52 
56 
13    O 
4 
8 
12 
16 
1320 
24 
28 
32 
36 
1340 
44 
48 
52 
56 
14    O 
4 
8 
12 
16 
1420 
24 
28 
32 
36 
1440 
44 
48 
52 
56 
15    0 
4 
8 
12 
16 
15  2O 
24 
28 
32 
36 
1540 
44 
48 
52 
56 
16    0 

9-736i 

7365 
7367 
7369 

0           I 

.     o  '.     i 

2 

2 

2 

0          I 
0           I 
I    j      2 
I 

I 
I 
I 
2 
2 

O          I 
0           I 

:     2 

2    j 
2    I 

0    1       I 
0    J       I 

!      2 

2 
2 

0          I 
O          I 
2 

2 
2 

0          1 

0          I 

2 

2 
2 

10.2639 
2637 
2635 
2633 
2631 

9.8125 
8128 
8131 

8i33 
8136 

, 

2 
2 

I 
2 
2 

2 
2 

I 

2 
2 

I 

2 
2 

I 

2 
2 

ff. 

10.1875 
1872 
1869 
1867 
1864 

10.0764 
0765 
0766 
0767 
0767 

9.9236 
9235 
9234 
9233 
9233 

48    O 
4756 
52 
48 
4744 
40 
36 
32 
28 
4724 
20 
16 
12 
8 
47    4 
47    0 
4656 
52 
48 
1644 
40 
36 
32 
28 
4624 
20 
16 
12 
8 
46    4 
46    O 
4556 
52 
48 
4544 
40 
36 
32 
28 
4524 
20 
16 
12 
8 
45    4 
45    0 
4456 
52 
48 
4444 
40 
36 
32 
28 
4424 
20 
16 
12 
8 
44    4 
44    O 

60 
59 
58  ! 
57 
56 
55 
54 
53 
52 
51 
5O 
49 
48 
47 
46 
45 
11 
43 
42  i 
41  1 
40 
39 
38  i 
37 
36 
35  ! 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 

8  ! 

6 

3 
2  i 
1 

9-737J 

7373 
7375 
7377 
7379 
9.7380 
7382 
7384 
7386 
7388 

10.2629 
2627 
2625 
2623 
2621 

9.8139 
8142 

8i45 
8147 
8150 

10.1861 
1858 
1855 
1853 
1850 

10.1847 
1844 
1842 

1839 
1836 

10.0768 
0769 
0770 
0771 
0771 

9.9232 
9231 
9230 
9229 
9229 

10.2620 
2618 
2616 
2614 
2612 

9.8153 
8156 
8158 
8161 
8164 

10.0772 
0773 
°774 
0775 
0776 

9.9228 
9227 
9226 
9225 
9224 

9-7390 
7392 
7394 
7396 
7398 

10.2610 
2608 
2606 
2604 
2602 

9.8167 
8169 
8172 

8i75 
8178 

10.1833 
1831 
1828 
1825 
1822 

10.0776 
0777 
0778 
0779 
0780 

9.9224 
9223 
9222 
9221 
9220 

9.7400 
7402 
7404 
7406 
7407 

10.2600 
2598 
2596 
2594 
^2593 
10.2591 
2589 
2587 
2585 
2583 

9.8180 
8183 
8186 
8189 
8191 

10.1820 
1817 
1814 
1811 
1809 

10.0781 
0781 
0782 

0783 
0784 

9.9219 
9219 
9218 

9217 
9216 

9.7409 
74" 
7413 
7415 
7417 

9.8194 

8i97 
8200 
8202 
8205 

I  O.I  806 

1803 
1800 
1798 
1795 

10.0785 
0786 
0786 
0787 
0788 

9.9215 
9214 
9214 
9213 
9212 
9.921! 
9210 
9209 
9209 
9208 

9.7419 

742i 
7423 
7425 
7427 

10.2581 
2579 
2577 
2575 

10.2572 
2570 
2568 
2566 
2564 
10.2562 
2560 
2558 

2551 
2549 
2547 

io72543~ 
2541 
2539 
2538 
2536 

9.8208 
8211 
8213 
8216 
8219 

10.1792 

1789 
1787 
1784 
1781 

10.0789 
0790 
0791 
0791 
0792 

9.7428 
7430 
7432 
7434 
7436 

9.8222 
8224 
8227 
8230 
8233 

10.1778 
1776 

1773 
1770 
176-7 

10.0793 
0794 

0795 
0796 
0796 

9.9207 
9206 
9205 
9204 
9204 

9-7438 
7440 

7442 
7444 
7445 

9-8235 
8238 
8241 

8243 
8246 

10.1765 
1762 

1757 
1754 

10.0797 
0798 
0799 
0800 
0801 
10.0802 
0802 
0803 
0804 
0805 

9.9203 
9202 
9201 
9200 
9199 

9-7447 
7449 
7451 
7453 
7455 

9.8249 
8252 
8254 
8257 
82*60 

IO.I75J 
1748 
1746 

1743 
1740 

9.9198 
9198 
9197 
9196 

9-7457 
7459 
7461 
7462 
7464 

9.8263 
8265 
8268 
8271 
8274 

10.1737 
1735 
1732 
1729 
1726 

10.0806 
0807 
0807 
0808 
0809 

9.9194 
9193 
9193 
9192 

9^190 
9189 
9188 

9187 
9187 

9.9186 

9.7466 
7468 
7470 
7472 
7474 
9.7476 

10.2534 
2532 
2530 
2528 
2526 
10.2524 

9.8276 

8279 
8282 
8284 
8287 
9.8290 

10.1724 
1721 
1718 
1716 
1713 
10.1710 

10.0810 
0811 
0812 
0813 
0813 
10.0814 

Cosec. 

' 

m    s 

Cos. 

O'.l     1s 

Sec. 

Cot. 

O'.l 

Di 

Tan. 

Sin. 

m    s 

Diff. 

123°  =  8h  12™  ]                                                                                                    [  3"  44m  =  56° 

TABLE  IX. 


61 


34° 

rrr  2"    16 

m     s 

16    O 
4 
8 
12 
16 
16  2O 
24 
28 
32 
36 
16  4O 
44 
48 
52 
56 
17    0 
4 
8 
12 
16 
17  2O 
24 
28 
32 
36 
1740 
44 
48 
52 
56 
18    0 
4 
8 
12 
16 
182O 
24 
28 
32 
36 
1840 
44 
48 
52 
56 
19    0 
4 
8 
12 
16 
1920 
24 
28 
32 
36 
1940 
44 
48 
52 
56 
20    0 

"] 

" 
Sin. 

9.7476 
7477 
7479 
7481 
7483 

9-7485 
7487 

7489 
749i 
7492 

I-OJf.  MlM'S. 

Diff. 
Cosec. 
O'.l     1"  | 

Tang 

Tan. 

9.8290 

8293 
8295 
8298 
8301 

eiits,  an 

D1.T. 
O'.lll8 

id  Sec; 
Cot. 

tints. 

Sec. 

10.0814 
0815 
0816 
0817 
0818 

[9" 
Cos. 

4O'»  =*  145° 

m     s    i     ' 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

/ 

o 

0 

2 
2 

O 
O 

I 
1 

; 
I 

2 

0 
O 

2 

2 

O 
0 

2 

2 

O 
O 

I 

I 

I 
2 
2 

O 
0 

2 
2 

i 
i 

i 
i 

2 

I 
I 

2 

I 
I 
2 

I 

I 
2 

I 

2 

10.2524 

2523 
2521 
25*9 

25r7 

o 
i 

i 
i 
i 

2 
2 

2 

2 

O 

2 
2 

2 
2 

O 

I 

2 
2 
2 
2 

0 

I 
I 
I 

2 
2 
2 
2 

O 

2. 
2 
2 
2 

0 
I 
I 
I 
I 
2 
2 
2 
2 

i 

2 
2 

I 

2 
2 

I 
2 
2 

I 

2 

2 

I 

2 
2 

I 
2 
2 

10.1710 
1707 

1705 
1702 
1699 

9.9186 
9185 
9184 
9183 
9182 

44    O 
4356 
52 
48 
4344 
40 
36 
32 
28 
4324 
20 
16 
12 
8 
43    4 
43    0 
4256 
52 
48 
4244 
40 
36 
32 
28 
4224 
20 
16 
12 
8 
42    4 
42    0 
4156 
52 
48 
4144 
40 
36 
32 
28 
4124 
20 
16 
12 
8 
41    4 
41    0 
4O56 
52 
48 
4O44 
40 
36 
32 
28 
4024 
20 
16 
12 
8 
40    4 
40    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 

28  ; 
27  ! 
26  i 
25  \ 
24 
23  ! 
22 
21 
20  ; 
19 
18 
17 
16 
15  i 
14  ! 
13 
12 
11 
1O 
9 
8 
7  : 
6 
5  • 
4 
3 
2 
1 
0 

10.2515 

2513 
2511 
2509 
2508 

9-8303 
8306 

8309 
8312 
83H 

10.1697 
1694 
1691 
1688 
1686 

10.0819 
0819 
0820 
0821 
0822 

9.9181 
9181 
9180 
9179 
9178 

9-7494 
7496 
7498 
7500 
7502 

10.2506 
2504 
2502 
2500 
2498 

9-831  7 
8320 

8323 
8325 
8328 

10.1683 
1680 
1677 

1675 
1672 

10.0823 
0824 
0825 
0825 
0826 

10.0827 
0828 
0829 
0830 
0831 

9.9177 
9176 
9175 
9175 
9174 

9-7504 
7505 
7507 
7509 
75" 
9-7513 
7515 
75i7 
75i8 
7520 

10.2496 
2495 
2493 
2491 

2i8_9_ 
10.2487 

2485 

2483 
2482 
2480 

9-8331 
8333 
8336 
8339 
8342 

10.1669 
1667 
1664 
1661 
1658 

9-9173 
9172 
9171 
9170 
9169 

9-8344 
8347 
8350 
8352 
8355 

10.1656 

1653 
1650 
1648 
1645 

10.0831 
0832 
o8J3 
0834 
0835 

9.9169 
9168 

9i67 
9166 

9165 

9.7522 

7524 
7526 
7528 
7529 

10.2478 
2476 

2474 
2472 
2471 

9.8358 
8361 

8363 
8366 
8369 

10.1642 
1639 
1637 

1634 
1631 

10.0836 

0837 
0837 
0838 
0839 

9.9164 
9163 
9163 
9162 
9161 

9-7531 
7533 

7535 
7537 
7539 

10.2469 
2467 
2465 
2463 
2461 

9.8371 
8374 
8377 

Us 

10.1629 
1626 
1623 
1621 
1618 

10.0840 
0841 
0842 

0843 
0844 

9.9160 
9159 
9158 
9157 
9156 

9-7540 
7542 
7544 
7546 
7548 

10.2460 
2458 
2456 
2454 
2452 

9i33l 
8390 

8393 
8396 

10.1615 
1612 
1610 
1607 
1604 

10.0844 
0845 
0846 
0847 
0848 

9.9156 
9155 
9154 
9153 
9152 

9-755° 
755i 

7553 
7555 
7557 

10.2450 

2449 
2447 

2445 
2443 

9.8398 
8401 
8404 
8406 
8409 

10.1602 
1599 
1596 
1594 
I59i 

10.0849 
0850 
0851 
0851 
0852 

9-9I51 
915° 
9149 
9149 
9148 

9-7559 
756i 
7562 

7564 
7566 

10.2441 
2439 
2438 
2436 

2434 

9.8412 

8415 
8417 
8420 

_8423_ 
9.8425 
8428 
8431 
8433 
_84J6 

9-8439 
8442 
8444 
8447 
8450 

10.1588 
1585 
1583 
1580 

1577 

10.0853 
0854 
0855 
0856 
0857 

9.9147 
9146 

9H5 

9144 

9H3 

9.7568 
7570 
757i 
7573 
7575 

10.2432 
2430 
2429 
2427 

_?42i 

10.2423 

2421 
2420 
2428 

2426 

io.i575 
1572 
1569 
1567 
I564 

10.0858 
0858 
0859 
0860 
0861 

9.9142 
9142 
9141 
9140 
9139 
0.9138 

9137 
9136 

9135 
9135 

9-7577 
7579 
758o 
7582 
_7584 
9-7586 

10.1561 

1558 
1556 
1553 
*55Q 
10.1548 

10.0862 
0863 
0864 
0865 
0865 

10.2424 

i  9-8452 

10.0866 

9-9r34 

ni    s 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s    |     ' 

Diff. 

Diff. 

124°  =  §h  16m  ]                                                                                                    [  3h  4Om  =  55° 

62 


TABLE  IX. 


35J  =  21'  2€ 

m    s 

m  -j 

Sin. 

Log.  Sines,  Tangents, 

sind  Secants. 

L9» 
Cos. 

9-9I34 
9133 
9132 

9I31 
913° 
9.9129 
9128 
9127 
9127 
9126 

^9125 
9124 
9123 
9122 
9I2I 
9.9120 
9119 
9119 
9118 
_  9"7 
9.9116 

9"5 
9114 

9"3 
9112 

9.9111 
9110 
9110 
9109 
9108 

36'r  =  1 

m    s 

10     0 
3956 
52 

48 
3911 

10 

36 
32 

28 
39  21 

20 
16 
12 

8 
39    1 
39    0 
38  56 
52 
18 
3811 
10 
36 
32 
28 
3821 
20 
16 
12 
8 
38    1 
38    O 
3756 
52 
18 
3711 
10 
36 
32 
28 
3721 
20 
16 
12 
8 
37    1 
37    0 
3656 
52 
18 
3611 
10 
36 
32 
28 
3621 
20 
16 
12 
8 
36    1 
36    0 

l-l3 

/ 

60 
59 
58 
57 
56 
55 
51 
53 
52 
51 
50 
19 
18 
17 
16 
15 
44 
43 
42 
41 
10 
39 
38 
37 
36 
35 
31 
33 
32 
31 
30 
29  | 
28 
27 
26 
25 
21  ! 
23 
22 
21 
2O 
19 
18 
17 
16 
15 
11 
13 
12 
11 
10 
9 
8 
7 
6 
5 
1 
3 
2 
1 

0 

Diff. 

Cosec. 

' 

Tan. 

Diff. 

Cot. 

Sec. 

O'.l 

1* 

o 

i 
i 

o 

I 
I 

0 

I 

I 

0 

I 

0 

I 
I 

O'.l 

1« 

2 

2 

2 
2 

J 

2 
2 

I 

2 

I 
2 
2 

I 
2 
2 

f 

1 
2 
3 
1 
5 
6 
7 
8 
9 
10 
11 
12 
13 
11 
15 
16 
17 
18 
19 
20 
21 
22 
23 
21 
25 
26 
27 
28 
29 
30 
31 
32 
33 
31 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
51 
55 
56 
57 
58 
59 
60 

2O    O 
1 
8 
12 
16 
20  20 
21 
28 
32 
36 
2010 
11 
18 
52 
56 
21    O 
4 
8 
12 
16 
2120 
21 
28 
32 
36 
2110 
11 
18 
52 
56 
22    0 
1 
8 
12 
16 
2220 
21 
28 
32 
36 
22  1O 
11 
IN 
52 
56 
23    0 
4 
8 
12 
16 
2320 
24 
28 
32 
36 
23  4O 
44 
48 
52 
56 
24    0 

9.7586 
7588 
7590 
7591 
7593 
9-7595 
7597 
7599 
7600 
7602 
977604" 
7606 
7607 
7609 
7611 

o 
•  o 

2 
2 

O 
0 

2 
2 

0 
0 

• 
2 
0 

o 

2. 

0 
O 

2 

O 
0 

2 

10.2414 
2412 
2410 
2409 
2407 
10.2405 

2403 
2401 
2400 
_239S 
10.2396 
2394 
2393 
2391 
2389 

10.2387 

2385 
2384 
2582 
2380 
10.237.* 
2376 
2375 
2373 
_._23Z1 
10.2369 
2368 
2366 
2364 
2362 

9.8452 

8455 
8458 
8460 
8463 
978^66" 
8468 
8471 
8474 
8476 
9.8479 
8482 
8484 
8487 
8490 

o 
i 

i 

: 

i 

2 

0 
I 
I 
I 
I 
2 
2 

2 

0 
I 

1 

2 

\ 

I 

I 

2 
2 
2 
2 

0 

I 
I 
I 
I 
2 
2 
2 
2 

O 

I 
I 
I 

I 
2 
2 
2 
2 

10.1548 

1545 
1542 
1540 
1537 

i°-I534 
1532 
1529 
1526 

_J524 
10.1521 

1518 
1516 

•1513 
1510 

10.0866 
0867 
0868 
0869 
0870 
10.0871 
0872 
0873 
0873 
0874 
10.0875 
o8"6 
0877 
0878 
0879 

10.0880 
0881 
0881 
0882 
0883 
10.0884 
0885 
0886 
0887 
0888 
10.0889 
0890 
0890 
0891 
0892 

9-76I3 
76i5 
7616 
7618 

702  "> 

9-8493 
^495 
8498 
8501 
8503 

10.1507 

1505 
1502 
1499 
1497 

9.7622 
7624 
7625 
7627 
7629 

9.8506 
8509 
8511 
8514 
8517 
9.8519 
8522 

8525 
8527 
8530 

10.1494 
1491 
1489 
1486 
1483 
10.1481 
1478 
1475 
H73 
1470 

9-7631 
7632 

7634 

_ll 

9.7640 
7641 
7643 
7645 
7647 

10.2360 
2359 
2357 
2355 
2353 

9.8533 
8535 
8538 
8541 
8543 

10.1467 

1465 
1462 

H59 
H57 

10.0893 
0894 
0895 
0896 
0897 
10.0898 
0899 
0899 
0900 
0901 

9.9107 
9106 
9105 
9104 
9103 

9.7648 
7650 
7652 
7654 
7655 

10.2352 

2350 
2348 
2346 
2345 

9.8546 
8549 
8551 
8554 
8557 

10.1454 

H5i 

1449 
1446 
1443 

9.9102 
9101 
9101 
9100 
9099 

9-7657 

$? 

7662 
7664 

10.2343 
2341 
2339 
233* 
2336 

1$ 

8565 
8567 
8570 

10.1441 
H38 
J435 
M33 
1430 

10.0902 
0903 
0904 
0905 
0906 

9.9098 
9097 
9096 

9095 
9094 

9.7666 
7668 
7669 
7671 
7673 

10.2334 
2332 
2331 
2329 
2327 

9.8573 

?sH 

8581 

8583 

10.1427 

1425 
1422 
1419 
1417 

10.0907 
0908 
0909 
0909 
0910 

9.9093 
9092 
9091 
9091 
9090 

9-7675 
7676 
7678 
7680 
7682 

10.2325 

2324 
2322 
2320 
2318 

9.8586 

8589 
8591 
8594 
8597 

10.1414 
1411 
1409 
1406 
1403 

10.0911 
0912 
0913 
0914 
0915 

9.9089 
9088 
9087 
9086 
9085 

9.7683 
7685 
7687 
7689 
7690 

10.2317 
2315 
23'3 
2311 
2310 

9.8599 

8602 
8605 
8607 
8610 

10.1401 
1398 
1395 
1393 
1390 

10.0916 
0917 
0918 
0919 
0920 

9.9084 
9083 
9082 
9081 
9080 

9.7692 

10.2308 

9.8613 

10.1387 

10.0920 

9.9080 

'     i 

m    s 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l     1s 

Tan. 

Cosec. 

Sin. 

in     3 

/ 

Diff. 

Diff. 

125°  =  8h  2Om  ]                                                                                                     [  3U  36m  =  54° 

TABLE  IX. 


36 

=  2"  21 

m     s 

210 
4 

8 
12 

16 
2  1  21> 
21 
23 
31 
36 
2140 
11 
4§ 

if 

** 

2->     O 
1 
8 
12 
16 
2520 
21 
23 
32 
36 
25  40 
44 
48 
52 
56 
26    0 
4 
8 
12 
16 
26  2O 
24 
28 
32 
36 
26  40 
44 
48 
52 
56 
27    0 
4 
8 
IS 
16 
27  20 
24 
28 
32 
36 
274O 
44 
48 
52 
56 
28    O 

>  ]             L,og.  Sines,  Tangents,  and  Secants.         [  9h  32m  =  143° 

Difif. 

Sin-  i  oUT- 

Cosec. 

Tan. 

Di 
0.1 

0 

i 

i 

i 
i 

2 

2 

2 

O 
I 
I 

I 
2 
2 

ff. 

1  s 

Cot. 

Sec. 

Cos. 

in     s 

/ 

O 

1 
2 
3 
4 
5 
6 
7 

i 

9 

10 
11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
;  22 
23 
i  24 
25 
26 
27 
2§ 
29 
1  30 
31 
!  32 
!  33 
34 
i35 
36 
37 
38 
39 
4O 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
6O 

9.7692  . 
7694        o 
7696  i      o 
7^97 
7^99 
9-77°i 
77^3 
77  J4 
7700^ 
7708        2 

o 
i 
i 

o 
I 

o 

I 
I 

o 

0 

I 

0 

10.2308 
2306 
2304 

23°3 
2301 

10.2299 
2297 

229J 

22)4 
2292 

9.8613 
8615 
8618 
8621 
8623 
9.8620 
8629 
8631 
8034 
.  8637 
9-8339 
8642 
8644 
8647 
8050 

9.8,552 

8655 
8658 
8600 
865  1 

i 

2 

I 
I 
2 

I 

I 

I 
2 

I 

I 
2 

I 
I 

2 

10.1387 

& 

1379 
1377 

10.1374 
1371 
1369 
1365 

I364 

10.0920 
0921 
0922. 
0923 
0924 
10.0925 
0926 
0927 
0928 
0929 

9.9080 
9079 
9078 
9077 
9076 

9-9075 
9074 
9073 
9072 
9£7i_ 
9.9070 
9069 
9069 
9068 
9067 
9.9066 
9065 
9364 
9003 
9062 
9.9061 
9060 

9059 
9358 

9057 

36    O 
35  56 
52 
48 
3544 
40 
36 
32 
28 
3524 
20 
16 
12 
8 
35    4 
35    O 
3456 
52 
48 
3144 
40 
36 
32 
28 
3424 
20 
16 
12 
8 
34    4 
34    0 
3356 
52 
48 
3344 
40 
36 
32 
28 
33  24 
20 
16 
12 
8 
33    4 
33    O 
3256 
52 
48 
3244 
40 
36 
32 
28 
3224 
20 
16 
12 
8 
32    4 
32    0 

60 
59 
58 
57 
-»:> 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28  , 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18  i 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

0.7710 
7711 
7/13 
7715 
77^ 
9.7718 
7723 
7722 
7723 
7711 

9-77^7 
7728 

773° 
7732 
7734 

o 

0 

2 
O 

o 

2 
O 

o 

2 

0 
O 

2 

0 
O 

I0.229J 
2289 
2287 
22S5 
2284 
10.2282 
2280 
2278 
2277 
2275 

10.1361 

1358 
1356 
{353 
1350 
10.1348 

1345 
1342 
1340 

1337 

10.0930 
0931 

093  i 
0932 
09^ 
10.0934 

°935 
0936 

0937 
0938 

10.2273 
2272 
227O 
2263 

2266 

9.866J  j 
8568  i      o 
8671' 

8674  : 

8675  1 

!o.i334 
1332 
1329 
1326 
1324 

10.0939 
0940 
0941 
0942 
0943 

9-7735 
7737 
7739 
7740 
7742 

9-7744 
774^ 
7747 
7749 
775] 

10.2265 
2263 
2261 
2260 

2258 

10.2256 

2254 

2253 
2251 

2249 

9.8679 

8682 
.8684 
8687 
8589 
9.8692 
8595 
8697 
8700 

8703 

2 
2 
2 

O 

2 
2 
0 

2 

2 

O 

2 
2 
2 

10.1321 
1318 
1316 
1313 
13" 
10.1308 
1305 
1303 
1300 
1297 

10.0944 
0944 
0945 
0946 
0947 
10.0948 
0949 
0950 

095  i 
0952 

9.9056 
9056 
9055 
9054 
9053 
9.9052 
9051 
9050 
9049 
_994& 
9.9047 
9046 
9045 
9044 
9043 
9.9042 
9041 
9041 
9040 
9039 

9-7752 
7754 
7756 
7758 
7759 

10.2248 
2246 

2244 

2242 

2241 

9.8705 
8708 
8711 

8713 
8716 

10.1295 
1292 
1289 
1287 
1284 

10.0953 
0954 
0955 
0956 

0957 

9.7751 

77^3 
7754 
7766 
7788 
9.7769 
7771 
7773 
7774 
7776 

10.2239 

2237 

2236 

2234 
2232 

9.8718 
8721 
8724 
8726 
8729 
9.8732 
8734 
8737 
8740 
8742 

10.1282 
1279 
1276 
1274 
1271 

10.0958 
0959 
0959 
0960 
0961 

10.223-1 
2229 

2227 

2226 
2224 

10.1268 
1266 
1263 
1260 
1258 
10.1255 

1253 
1250 
1247 

1245 

10.0962 
0963 
0964 
0965 
0966 

9.9038 

9037 
9036 

9035 
9°34 

9.7778 
7780 

7781 
7783 
7785 

10.2222 
.   2220 
2219 
2217 
2215 

9-8745 
8747 
8750 

8753 
8755 

10.0967 
0968 
0969 
0970 
0971 

9-9033 
9032 
9031 
9030 
9029 
9.9028 
9027 
9026 
9025 
9024 

9-7786 
7788; 
7790! 
779i  | 
7793  !      2 

IO.22I4 
2212 

2210 
22O9 
22O7 
IO.22O5 

9.8758 
8761 

8763 
8766 
8769 

10.1242 
1239 
1237 

!234 

1231 

10.0972 

0973 
0974 

0975 
0976 

9-7795  i 

9.8771 

10.1229 

10.0977 

9.9023 

/ 

in    s 

Cos. 

0.1 

1s 

r-  Bso- 

Cot. 

O'.l 

1s 

Tan. 

Cossc. 

Sin. 

m    s 

/ 

Di 

Difif. 

!  126°  =  81'  24m  ]                                                                                                    [  3b  32m  =  53°  i 

64 


TABLE  IX. 


37D  =  2  '  28  u  ]            JLogr.  Sines,  Tangents,  smtl  Secants.         [  91'  28m  =  142° 

' 

m    s 

Diff. 
Sin. 
O'.l    1 

Cosec. 

Tan. 

Diif. 
O'.'l     1 

0           I 

i 

2    i 
2 
2    ! 

0           I 

I       I 

I           2 
I 

I 

I 
2 
2 
2 

O           I 

I 
I 

2   4 

2 

0          I 
I           i 
I           2 
I 

I 

2 
2 
2 

Oil 
I 

2 

1 

2 

2    \ 
2 

0          I 
I 

2 

2 
2 
2 

Cot. 

Sec. 

Cos.  S 

j 

9.9323! 

9  3.:  1  1 
932  >1 

9318 

9317 

9016 

9015 
9-9314; 

9312 
9011 

9010 

9.9337 
9008 
9307- 
9306 
9005 

ni    s    ; 

3-2     <&  . 
31  33 
5  £ 
49 

31  11 

JIO 

32 
28 
3124 

16 
12 

8 
31     4 

31     0 

:^;>| 

48 

3O44 
40 

36 
32 
28 
30  24 
20 
16 
12 
8 
'  3O    4 
30    0 
2956 
52 
48 
2944 
4O 
36 
32 
28 
29  24 
20 
16 
12 
8 
29    4 
29    O 
28  56 
52 
48 
2844 
4O 
36 
32 
28 
2W  2  1 
20 
16 
12 
8 
28    4 
28    O 

60 
39 

57 

53 
51 
53 
52 
51 

50 
49 
48 
47 
16 
45 
44 
43 
42 
41 

40 

39 
38 
37 
36 
35 
34 

32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

0 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
21 
25 
26 
27 
'28 
29 
30 
31 
32 
33 
34 
35 
36 
i  37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

28    0 
4 
8 
12 
16 
28  2O 
21 
28 
32 
36 

9-7795 
7790 

7793 
7833 
7831 

0           0 
O           I 

I 

2 

0          0 
0          I 

J    ' 

I 

I 
I 
I  i 

2 
0   j      0 

o  i     i 

I  i 

2    ' 

O    ;      O 
0           I 
I    j       I 
I    j 
I    | 
I 
I 
I 
2 

0          0 
O          I 

2 

0   j      0 
O           I 

I       I 
I 
I 

I 
I 

2 

10.2205 
2204 

2202 

2203 
2  I  Q  1 

9.8771 

8774 
8770 

8779 
878-5 

10.1229 

122  ) 
1224 
122( 

1218 

10.0977 

0977 
0970 

9.7803 

7805 
7800 
7808 
7810 

10.2197 

2195 
2194 
2192 
2191 

9.8784 

8787 
8793 
8792 
8795 
9.879; 

8333 
8833 
8805 
8338 

1O.I2IO 
1213 
I2IO 
I2O8 
1  20", 

10.1203 
I2O3 

"97 
"95 
1192 

10.0961 
0982 
0983 
0984 
0985 

2840 
44 

48 
52 
56 
29    0 
4 
8 
12 
16 
29  2O 
24 
28 
32 
36 
2940 
41 
48. 
52 
53 
3$    O 
4 
8 
12 
16 
3O2O 
24 
28 
32 
36 
3O4O 
44 
48 
52 
56 
31     O 
4 
8 
12 
16 
3120 
24 
28 
32 
36 
3140 
44 
48 
52 
56 
32    0 

9.7811 
7813 
7815 
7810 
7818 
9.7820 
7821 
7823 
7825 
7820 

I0.2I89 
2187 
2185 
2184 
2182 

10.0980 
0987 
0988 
0989 
0990 

10.2180 

2179 
217; 
2175 

9.88II 
8813 
8810 
8318 
8821 
9.8.324 
•      8826 

8329 
8831 
8834 

10.1189 
1187 
1184 
1182 
ii79 
10.1170 
"74 

1169 
1166 

10.0991 
0992 

0993 
0994 

0991 

10.0990 
0997 
0998 
0999 

IO33 
10.1000 
IOOI 
IOO2 
1003 
103-1- 
10.1005 
IOOO 

1007 
1008 
1009 

9.7828 
7830 
7831 
7833 
7835 

10.2172 
2170 
2169 
2167 
2165 
10.2164 
2162 
2160 

2159 
2157 

10.2150 

2154 
2152 
2151 
2149 

9.9004 

9033 

9032 

9001 
9000 

9.9000 

8999 
8998 
8997 
8996 

9.8995 
8994 

8993 
8992 
8991 

9.7830 
7838 
7840 
7841 
7_^43 
9.7844 
7840 
7848 
7849 
7851 

9.8837 

8839 
8842 

8^45 
•  8847 
9.8850 
8852 

8855 
8858 
8860 

10.1163 
1161 
1158 
"55 
"53 
10.1150 
1148 

"45 
1  142 
1140 

9'7JJ54 

7858 
7859 

10.2147 
2146 

2144 
2142 
2141 

8868 
8871 
8873 

10.1137 
"35 
"32 
1129 
1127 

IO.IOIO 

KM  I 
IOI2 
1013 
IOI4 

9.8990 
8989 
8988 

8987 

8986 

9.7861 
7803 
7804 
7855 
7867 

10.2139 
2137 

2134 

9.8876 
8879 
8881 
8884 
8886 

9.8889 
8892 
8894 
8897 
8899 
9.8902 
8905 
8907 
8910 
8912 

10.1124 

II2I 
III9 

1116 
1114 

IO.IOI5 

1016 
1017 
1018 

IOIQ 

9.8985 
8984 
8983 

8982 
8981 

9.7869 
7871 
7872 

7874 
7870 

10.2131 
2129 

2128 
2126 
2124 

10.2123 

2121 
2120 

2118 
2116 

IO.IIII 

1108 
1106 
1103 

IIOI 

IO.IO2O 
IO2I 
IO22 
1023 
1024 

9.8980 
8979 
8978 

8977 
8976 

9-7877 
7879 
7880 
7882 
7880 

10.1098 
1095 
1093 

IO.IO25 
1026 
IO27 
1028 
1029 

9-8975 
8974 
8973 
8972 

8971 

9.7885 
7887 
7889 
7890 
7892 

10.2115 

2113 

2III 

2IIO 
2108 

9.8915 

8918 
8920 

8923 
8925 

10.1085 
1082 
1080 
1077 
1075 

10.1030 
1031 
IO32 

1034 

9.8970 
8969 
8968 
8967 
8966 

9-7893 
Cos. 

10.2107 

9.8928 

10.1072 

IO.IO35 

9.8965 

m     s 

O'.l     1" 

Sec. 

Cot. 

O'.l    1s 

Diff 

Tan. 

CDSCC. 

Sin. 

m    s 

' 

Diff. 

1  21?'  —  K'>  £QI"  1                                                                                                                                f  3  '  28!"  — 

K  4T  •       •           V        *  9   •     |                                                                                                                                                                                                         L 

52 

TABLE  IX. 


65 


38°     2"  32>u  j            Log.  Sines,  Tangents,  and  Secants.         [  9h  24'"  =  141°  | 

' 

m    B 

Sin. 

Diff. 

Coaec. 

Tan. 

Dii 
O'.l 

f. 

Cot. 

Sec. 

10.1035 
1036 

1037 
1038 

10.1040 
1041 
1042 

1043 
1044 

Cos. 

m     s         ' 

O'.l 

1s 

O 

1 
2 
3 
4 
5 
6 

8 
9 
1O 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
i  38 
39 
40 
141 
|42 
43 
44 
45 
46 
47 
48 
49 
5O 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

32    O 
4 

8 
12 
16 
3220 
24 
28 
32 
36 
32  4O 
44 
48' 
52 
56 
33    0 
4 
8 
12 
16 
33  2O 
24 
28 
32 
36 
33  4O 
44 
48 
52 
56 
34    O 
4 
8 
12 
16 
3420 
24 
28 
32 
36 
34  4O 
44 
48 
52 
56 
35    0 
4 
8 
12 
16 
3520 
24 
28 
32 
36 
3540 
44 
48 
52 
56 
36    O 

9.7893 
7895 
7897 
7898 

9.7901 
7903 
7905 
7906 

7908 
9.7910 
7911 
7913 
79*4 

"9^7918" 
7919 
7921 
7922 
7924 
9.7926 

7927 
7929 

793° 

9-7934 
7935 
7937 
7938 
7940 

0 

o 

2 

O 
O 

2 

0 
0 

2 
O 

o 

2 

O 
0 

2 

O 
O 

I 

2 

o 
i 
i 

0 

o 
I 
I 

o 
I 

I 

o 

o 
I 

10.2107 
2105 
2103 

2102 
2IOO 
10.2099 
2097 
2095 
2094 
2092 

9.8928 
8931 
8933 
8936 

9.8941 

8944 
8946 

8949 
8952 

o 

2 
2 

2 

O 

I 
I 
I 
I 
I 
2 
2 
2 

O 

2 

2 

O 

2 
2 
2 

O 

2 
2 

2 

O 

2 

2 
2 

i 
I 

2 

2 

I 
I 

2 

I 
I 

2 

2 

I 

I 
2 

10.1072 
1069 
1067 
1064 
1061 
10.1059 
1056 
1054 
1051 
1048 

9.8965 
8964 
8963 
8962 
8961 
9.8960 

8959 
8958 

f95! 
8956 

9-8955 
8954 
8953 
8952 

8951 

28    0  j 
27  56 
52 

48 
27  44 

to 

36 
32 
28 
2724 
2O 
16 
12 
8 
27    4 
27    0 
2656 
52 
48 
2644 
40 
36 
32 
28 
2624 
2O 
16 
12 
8 
26    4 
26    0 
25  56 
52 
48 
2544 
40 
36 
32 
28 
2524 
20 
16 
12 
8 
25    4 
25    0 
24  56 
52 
48 
2444 
40 
36 
32 
28 
2424 
2O 
16 
12 
8 
24    4 
24    0 

111      S 

6O 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50  \ 
49 
48  \ 
47  i 
46  i 
45  i 
44 
43 
42 
41 
40 
39 
38 
37 
36  i 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19  I 
18 
17 
16 
15 
14 
13 
12  I 
11 
10 
9 
8 
7 
6 
5 
I  \ 
3 
2 
1  . 

10.2090 
2089 
2087 
2086 
2084 
10.2082 
2081 
2079 
2078 
2076 

9-8954 
8957 
8959 
8962 
8965 

10.1046 

1043 
1041 
1038 

10.1033 
1030 
1028 
1025 

1022 
IO.IO2O 
IOI7 
1015 
IOI2 
IOIO 

10.1007 
IOO4 
IOO2 

0999 

10.0994 
0991 
0989 
0986 
0984 

10.0981 
0978 
0976 

0973 
0971 

10.1045 
1046 
1047 
1048 
1049 

9.8967 
8970 
8972 
8975 
8978 
9.8980 
8983 

8985 
8988 
__§99_o 
9.8993 
8996 
8998 
9001 
9003 

10.1050 
1051 
1052 

1053 
_1°54 
10.1055 
1056 

1057 

1058 

_  I059_ 
i  o.i  060 
1061 
1062 
1063 
1064 

9.8950 

8949 
8948 

8947 
8945 

9.8945 
8944 
8943 

8941 
9.8940 

8939 
8938 

8937 
8930 

IO.2O74 
2073 
2071 
2O7O 
2068 
IO.2O66 
2065 
2063 
2062 
2O6O 

9.7941 
7943 
7945 
7946 

9-7949 
795  i 
7953 
7954 
7956 

IO.2O59 
2057 

2055 
2054 
2052 

10.2051 
2049 
2047 
2046 
2044 

9.9006 
9009 
9011 
9014 
9016 
9.9019 
9022 
9024 
9027 
9029 

10.1065 
1066 
1067 
1068 
1069 

10.1070 
1071 
1072 
1073 
1074 

9-8935 
8934 
8933 

_8£31 

9.8930 
8929 
8928 
8927 
8926 

9-7957 
7959 
7960 
7962 
7964 

10.2043 
2041 
2040 
2038 
2036 

9.9032 
9035 
9°37 
9040 
9042 

10.0968 
0965 
0963 
0960 
0958 

10.1075 
1076 
1077 
1078 
1079 

9.8925 
8924 

8923 
8922 
8921 

0.7965 
7967 
7968 
7970 
7972 

10.2035 
2033 
2032 
2030 
2028 

9.9045 
9047 
9050 
9053 
9055 

10.0955 

0953 
0950 
0947 
0945 

10.1080 
1081 
1082 
1083 
1084 

9.8920 
8919 
8918 

8917 
8916 

9.8915 
8914 

8913 
8912 
8911 

9-7973 

7975 
7976 

7978 
7979 

IO.2O27 
2025 
2O24 
2022 
2O2I 

9.9058 
9060 
9063 
9066 
9068 

10.0942 
0940 
°937 
0934 
0932 

10.1085 
1086 
1087 
1088 
1089 

9-798i 
7982 
7984 
7986 
7987 

IO.2OI9 
20l8 
2Ol6 
2OI4 
2013 
IO.2OII 

9.9071 

9073 
9076 
9079 
9081 

10.0929 
0927 
0924 
0921 
0919 

10.1090 
1091 
1092 
1093 
1094 

9.8910 
8909 
8908 
8907 
8906 

9.7989 

9.9084 

10.0916 

10.1095 

9.8905 

' 

m    B 

Cos. 

O'.l 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

/ 

Diff. 

Diff. 

128°  =  8h  32m  ]                                                                                                 [  3h  24m  =  51°  j 

66 


TABLE  IX. 


39D 

O 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
;  24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

=  2h  36™]              Log.  Si  nos,  Tangents,  and  Secants.         [911  2Om  =  1  1O 

m    3 

Sin. 

Diflf. 

Cosec. 

Tan. 

Diflf. 

Cot. 

I 

Sec.  |  Cos.  1  m     s 

O'.l 

I8 

O'.l 

I8 

36    O 
4 

8 
12 
16 
36  2O 
24 
28 
32 
36 
3640 
44 
48 
52 
56 
37    O 
4 
8 
12 
16 
3720 
24 
28 
32 
36 
3740 
-44 
48 
52 
56 
38    0 
4 
8 
12 
16 
3820 
24 
28 
32 
36 
3840 
44 
48 
52 
56 
39    0 
4 
8 
12 
16 
39  2O 
24 
28 
32 
36 
39  4O 
44 
48 
52 
56 
40    0 

9.7989 
7990 
7992 
7993 
7995 

o 
o 

0 

I 

I 
I 
I 
I 
I 

o 

0 

0 

o 
o 
o 

I 

o 
o 

0 
I 

0 

o 
o 
I 

o 
o 

0 

I 

o 
I 

I 

o 
I 

I 

, 
0 

I 

o 
I 
I 

0 

I 

I 

0 

I 
I 

IO.2OII 
2OIO 
2008 
2OO7 
2OO5 

9.9084 
9086 
9089 
9091 
9094 

0 

o 
I 

I 
I 
I 

2 
2 
2 

0 
O 

1 
I 
2 
2 

2 

O 
O 

I 
I 
I 

2 

2 

O 
O 

I 

I 

2 

O 
O 

I 
I 
2 
2 
2 

O 
O 

I 

2 
2 
2 

i 
i 

2 

I 
jj 

i 

2 

I 
I 

2 

I 
I 

2 

10.0916 
0914 
0911 
0909 
0906 

10.1095 
1096 
1097 
1098 
1099 

9.8905 
8904 
8903 
8902 
8901 

24    0 
23  56 
52 
48 
2344 
40 
36 
32 
28 
2324 
20 
.      16 
12 
8 
23    4 
23    O 
22  56 
52 
48 
2244 
40 
36 
32 
28 
2224 
20 
16 
12 
8 
22    4 
22    0 
2156 
52 
48 
2144 
40 
36 
32 
28 
21  24 
06 
16 
12 
8 
21    4 
21    O 
2056 
52 
48 
2044 
40 
36 
32 
28 
2024 
20 
16 
12 
8 
20    4 
20    0 

60 
59 
58 
57 
56 
55 
54 
53 
52  1 
31 
5O 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38  i 
37 
36 
35 
34 
33 
32 
31 
3O 
29 
28  1 
27  ! 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

• 

9.7997 
7998 
8000 
8001 
8003 

10.2003 
2002 
2OOO 
1999 
1997 

9.9097 
9099 
9102 
9104 
9107 

10.0903 
0901 
0898 
0896 
0893 

10.  1  100 
IIOI 
IIO2 
1103 
1104 

9.8900 
8899 
8898 
8897 
8896 

9:8895 

8894 
8893 
8892 
8891 

9.8004 
8006 
8007 
8009 
8010 

10.1996 
1994 

1993 
1991 
1990 

9.9110 
9112 

9H5 
9117 

9120 

10.0890 
0888 
0885 
0883 
0880 

10.1105 

1106 
1107 

1  105 

1109 

9.8012 
8014 
8015 
8017 
8018 

10.1988 
1986 
1985 
1983 
1982 

9.9122 
9125 
9128 
9130 
9133 

10.0878 

0875 
0872 
0870 
0867 

IO.IIIO 

IIII 

1112 
III3 
."15 

9.8890 
8889 
8888 
8887 
8885 

9.8020 
8021 
8023 
8024 
8026 

10.1980 
1979 

1977 
I976 

J974 

9-9135 
9138 
9140 

9H3 
9146 

10.0865 
0862 
0860 
0857 
0854 

IO.III6 
III7 

1118 
1119 

II2O 

q.8884 
"  8883 
8882 
8881 
8880 

9.8027 
8029 
8031 
8032 
8034 

10.1973 
1971 
1969 
1968 
1966 

9.9148 
9151 
9153 
9150 
9158 

10.0852 
0849 
0847 
0844 
0842 

IO.II2I 
1  122 
II23 
II24 
"25 

9.8879 
8878 

8877 
8876 

8875 

9-8035 
8037 
8038 
8040 
8041 

10.1965 
1963 
1962 
1960 
1959 

9.9161 
9164 
9166 
9169 
9171 

10.0839 
0836 
0834 
0831 
'  0829 

IO.II26 
1127 
1128 
1129 
1130 

9.8874 

8873 
8872 
8871 
8870 

9.8043 
8044 
8046 

8047 
8049 

10.1957 
1956 
1954 
1953 
I951 

9.9174 
9176 
9179 
9182 
9184 

10.0826 
0824 
0821 
0818 
0816 

IO.II3I 
II32 

"33 
"34 
"35 

9.8869 

8868 
8867 
8866 

8865 

9.8050 
8052 

8053 
8055 
8056 

10.1950 
1948 
1947 
1945 
1944 

9.9187 
9189 
9192 
9194 
9197 

10.0813 
0811 
0808 
0806 
0803 

10.1136 

"37 
1138 
1140 
1141 

9.8864 
8863 
8862 
8860 
8859 

9.8058 
8060 
8061 
8063 
8064 

10.1942 
1940 
'939 
1937 
1936 

9.9200 
9202 
9205 
.9207 
9210 

10.0800 
0798 
0795 
0793 
0790 

10.1142 
"43 
"44 
"45 
1146 

9.8858 
8857 
8856 
8855 
8854 

9.8066 
8067 
8069 
8070 
8072 
9.8073 
8075 
8076 
8078 
8079 

10.1934 
1933 
I93i 
1930 
1928 

9.9212 

9215 
9218 
9220 
9223 
9.9225 
9228 
9230 

9233 
9236 
9.9238 

10.0788 
0785 
0782 
0780 
0777 

10.1147 

1148 

"49 
1150 
1151 

10.1152 
"53 
"54 
"55 
1156 

9-8853 
8852 
8851 
8850 
8849 

10.1927 

1925 
1924 
1922 
1921 

10.0775 
0772 
0770 
0767 
0764 

9.8848 
8847 
8846 
8845 
8844 

9.8843 

9.8081 

10.1919 

10.0762 

0.1157 

/ 

m   B 

Cos. 

O'.l 

1" 

Sec. 

Cot. 

O'.l 

I8 

Tan. 

Cosec. 

Sin. 

in     s 

/ 

Diflf. 

Diflf. 

129°'=  8h  36m  ]                                                                                                 [  8h  2O'n  =  5O° 

TABLE  IX. 


67 


1O  -  2h  4Om  ]            L.ogr.  Sines,  Tangents,  and  Secants.        [  9h  16m  =  139° 

/ 

m     s 

Sin. 

Diff. 

Cosec. 

Tan. 

Di 
CK1 

S. 

Cot. 

Sec. 

Cos. 

m    s 

/ 

O'.l 

I8 

o 
I 

o 
I 

0 

I 

I 

o 
I 

I 

o 
I 

I 

0 

I 

I 

1s 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 

15 
16 
17 
18 
19 
2O 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
144 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

/ 

4O    O 
4 

8 
12 
16 
4O2O 
24 
-    28 
32 
36 
4040 
44 
48 
52 
56 
41    0 
4 
8 
12 
16 
41  2O 
24 
28 
32 
36 
4140 
44 
48 
52 
56 
42    0 
4 
8 
12 
16 
4220 
24 
28 
32 
36 
4240 
44 
48 
52 
56 
43    0 
4 
8 
12 
16 
43  2O 
24 
28 
32 
36 
4340 
44 
48 
52 
56 
44    0 

9.8081 
8082 
8084 
8085 
8087 

0 

o 
o 

0 

o 

0 

I 

0 

o 
o 

0 

o 

0 

I 

0 

o 
o 
I 

0 

o 

0 

10.1919 
1918 
1916 
1915 
1913 

9.9238 
9241 

9243 
9246 
9248 

o 
o 

i 
i 
i 
i 

2 
2 
2 

0 
O 
I 

I 
I 
I 
2 
2 
2 

0 
O 
I 
I 

2 
2 
2 

0 
O 
I 
I 

I 
I 
2 
2 
2 

0 
O 
I 
I 
I 
I 
2 
2 
2 

O 
0 
I 
I 

I 
I 
2 
2 
2 

i 

2 

I 

2 

I 
I 
2 

I 

2 

2 

I 
2 

10.0762 
0759 
0757 
0754 
0752 

10.1157 

H59 

1160 
1161 
1162 

9:8843 
8841 
8840 
8839 
8838 

"9^8837 
8836 

8835 
8834 

8833 

20    0 
1956 
52 
48 
1944 
40 
36 
32 
28 
1924 
20 
16 
12 
8 
19    4 
19    0 
1856 
52 
48 
1844 
4O 
36 
32 
28 
1824 
20 
16 
12 
8 
18    4 
18    0 
1756 
52 
48 
1744 
40 
36 
32 
28 
1724 
20 
16 
12 
8 
17    4 
17    0 
1656 
52 
48 
1644 
40 
36 
32 
28 
1624 
20 
16 
12 
8 
16    4 
16    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 

30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 

0 

9.8088 
8090 
8091 

8093 
8094 

10.1912 
1910 
1909 
1907 
1906 

9.9251 

9254 
9256 

9259 
9261 

10.0749 
0746 
0744 
0741 
Q739 
10.0736 
0734 
0731 
0729 
0726 

10.1163 
1164 
1165 
1166 
1167 

9.8096 
8097 
8099 
8100 
8102 

10.1904 

1903 
1901 
1900 
1898 

9.9264 
9266 
9269 
9271 
9274 

10.1168 
1169 
1170 
1171 
1172 

9.8832 
8831 
8830 
8829 
8828 

9.8103 
8105 
8106 
8108 
8109 

10.1897 

1895 
1894 
1892 
189i 

9.9277 
9279 
9282 
9284 
9287 

10.0723 
0721 
0718 
0716 
0713 

10.1173 

H75 
1176 
1177 

1178 

9.8827 
8825 
8824 
8823 
8822 

9.8111 
8112 
8114 
8115 
8117 

10.1889 
1888 
1886 
1885 
1883 

9.9289 
9292 

9295 
9297 
9300 

10.0711 
0708 
0705 
0703 
0700 

10.1179 

1180 
1181 

1182 
1183 

9.8821 
8820 
8819 
8818 
8817 

9.8118 
8120 
'8121 
8122 
8124 

10.1882 
1880 

1879 
1878 
1876 

9.9302 
9305 
93°  7 
9310 
9312 

10.0698 
0695 
0693 
0690 
0688 

10.1184 
1185 
1186 
1187 
1188 

9.8816 
8815 
8814 
8813 
8812 

9.8125 
8127 
8128 
8130 
8131 

10.1875 

1873 

1872 
1870 
1869 

9-93I5 
93i8 
9320 
9323 
9325 

10.0685 
0682 
0680 
0677 
0675 

10.1190 
1191 
1192 
"93 
"94 

9.8810 
8809 
8808 
8807 
8806 

9.8i33 
8134 
8136 

8i37 
8i39 

10.1867 
1866 
1864 
1863 
1861 

9-9328 
9330 
9333 
9335 
9338 

9-9341 
9343 
9346 
9348 
935i 

10.0672 
0670 
0667 
0665 
0662 

10.1195 
1196 
1197 
1198 
1199 

9.8805 
8804 
8803 
8802 
8801 

9.8140 
8142 
8143 
8i45 
8146 

I  O.I  860 

1858 

1857 
1855 

1854 

10.0659 
0657 
0654 
0652 
0649 

10.1200 
1201 
1203 
1204 

I2O5 

9.8800 
8799 
8797 
8796 
8795 
9.8794 

8793 
8792 

8791 
8790 

9.8148 

8149 
8150 
8152 
8i53 

10.1852 
1851 
1850 
1848 
1847 

9-9353 
9356 
9358 
9361 

9364 

10.0647 
0644 
0642 

<*>39 
0636 

10.1206 
1207 
1208 
1209 
I2IO 

9.8i55 
8156 
8158 
8159 
8161 

10.1845 
1844 
1842 
1841 
1839 

9.9366 
9369 
937i 
9374 
9376 

10.0634 
0631 
0629 
0626 
0624 

IO.I2II 
1212 
1213 
1215 

1216 

9.8789 
8788 
8787 
8785 
8784 

9.8162 
8164 
8165 
8167 
8168 

10.1838 
1836 
1835 
1833 
1832 

9-9379 
938i 
9384 
9387 
9389 

10.0621 
0619 
0616 
0613 
.  0611 

10.1217 

1218 
1219 

1220 
1221 

9.8783 
8782 
8781 
8780 
8779 

9.8169 

10.1831 

9.9392 

10.0608 

10.1222 

9.8778 

m    a 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m     s 

/ 

Diff. 

Diff. 

13O°  =  8h  4Om  ]                                                                                                    [  3h  16m  =  49° 

TABLE  IX. 


41°  =  2h  44m  ]           Log.  Sines,  Tangents,  and  Secants.         [  9h  12I1!  if  138= 

/ 

in    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec.      Cos. 

Ill      S 

6O 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48  : 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29  ; 

28  | 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 

i 

O'.l     I8 

0.1     1* 

0 
1 

2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
i  57 
1  58 
{  59 
60 

44    0 
4 

8 
12 
16 
4420 
24 
28 
32 
36 
4440 
44 
48 
52 
56 
45    0 
4 
8 
12 
16 
4520 
24 
28 
32 
36 
4540 
44 
48 
52 
56 
46    0 
4 
8 
12 
16 
4620 
24 
28 
32 
36 
4640 
44 
48 
52 
56 
47    0 
4 
8 
12 
16 
4720 
24 
28 
32 
36 
4740 
44 
48 
52 
56 
48    0 

9.8169 
8171 
8172 
8174 
8i75 

o 
o 

0 

I 
o 

0 

o 

o 
o 
o 

o 

0 
0 

I 

0 

o 
o 

0 

o 

0 

o 
1 

I 

o 

I 
I 

o 

I 
I 

o 
I 

I 

0 

I 
I 

o 
I 

10.1831 
1829 
1828 
1826 
1825 

9.9392 
9394 
9397 
9399 
9402 

o 
o 

I 

2 
2 
2 

O 
O 
I 

I 
I 

2 
2 
2 

0 
O 

I 

I 

2 
2 

2 

O 
O 
I 

I 
I 
2 
2 
2 

O 
0 
I 
I 

I 

2 
2 
2 

0 
O 

2 

2 

2 

2 

I 

2 

I 

I 
2 

I 
I 
2 

I 
I 
2 

I 
I 
2 

10.0608 
0606 
0603 
0601 
0598 

IO.I222 
I223 
1224 
1225 
1227 

9.8778 
8/77 
8776 

8775 
8773 
9.3772 
8771 
8770 
8769 
8768 

16    O 

1556 
52 
48 
1544 
40 
36 
32 
28 
1524 
20 
16 
12 
8 
15    4 
15    0 
1456 
52 
48 
1444 
40 
36 
32 
28 
1424 
20 
16 
12 
8 
4 
14    O 
1356 
52 
48 
1344 
40 
36 
32 
28 
1324 
20 
16 
12 
8 
13    4 
13    0 
1256 
52 
48 
1244 
40 
36 
32 
28 
1224 
2O 
16 
12 
8 
12    4 
12    0 

9.8177 
8178 
8180 
8181 
8182 

10.1823 
1822 
1820 
1819 
1818 

9.9404 
9407 
9409 
9412 
9415 

10.0596 

0593 
0591 
0588 
0585 

IO.I228 
1229 
1230 
I23I 
I232 

9.8184 
8185 
8187 
8188 
8190 

10.1816 
1815 
1813 
1812 
1810 

9-94I7 
9420 
9422 

9425 
9427 

10.0583 
0580 
0578 
0575 
0573 

IO.I233 
1234 
1235 
1237 

1238 

9.8767 
8766 
8765 
8763 

8762 

9.8191 

8i93 
8194 
8i95 
8i97 

10.1809 
1807 
1806 
1805 
1803 

9.9430 
9432 
9435 
943s 
944° 

10.0570 
0568 

0565 
0562 
0560 

10.1239 
I24O 
1241 
1242 
1243 

9.8761 
8760 
8759 
8758 
8757 

9.8198 
8200 
8201 
8203 
8204 

10.1802 
1800 
1799 
1797 
1796 

9-9443 
9445 
9448 
9450 
9453 

10-0557 
°555 
0552 
0550 
0547 

IO.I244 

1245 
1247 
1248 
1249 

9.8756 
8755 
8753 
8752 
875i 

9.8205 
8207 
8208 
8210 
8211 

10.1795 

*793 
1792 
1790 
1789 

9-9455 
9458 
9460 

9463 
9466 

10.0545 
0542 
0540 
0537 
0534 

IO.I25O 
1251 
1252 

1253 
-  1254 

9.8750 
8749 
874» 
8747 
8746 

9.8745 
8743 
8742 
8741 
8740 

9.8213 
8214 
8216 
8217 
8218 

10.1787 
1786 

1784 

17.83 
1782 

9.9468 
947i 
9473 
9476 

9478 

10.0532 
0529 

0527 
0524 
0522 

10.1255 
1257 
1258 
1259 
I26O 

9.8220 
8221 
•8223 
8224 
8225 

10.1780 
1779 

1777 
1776 

1775 

9.9481 

9483 
9486 
9488 
9491 

10.0519 
0517 
05H 
0512 
0509 

IO.I26I 
1262 
1263 
1264 
1266 

9ff* 
tg 

8734 

9.8227 
8228 
8230 
8231 
8233 

10.1773 
1772 
1770 
1769 
1767 

9-9494 
9496 

9499 
95oi 
95°4 

10.0506 
0504 
0501 
0499 
0496 
10.0494 
0491 
0489 
0486 
0484 

10.1267 
1268 
1269 
I27O 
1271 
IO.I272 
1273 
1275 
1276 
1277 

9-8733 
8732 
8731 
8730 
8729 

9.8234 
8235 
8237 
8238 
8240 

10.1766 
1765 
1763 
1762 
1760 

9.9506 
95°9 
95" 
95  H 
95i6 

9.8728 
8727 
8725 
8724 
8723 

9.8241 
8242 
8244 
8245 
8247 

10.1759 
1758 
1756 
1755 
1753 

9-95J9 
9522 

9524 
9527 
9529 

10.0481 
0478 
0476 

0473 
0471 

IO.I278 
1279 
1280 
I28l 
1282 

9.8722 
8721 
8720 
8719 
8718 

9.8248 
8249 
8251 
8252 
8254 

10.1752 
I75i 
1749 
1748 
1746 

9-9532 
9534 
9537 
9539 
9542 

10.0468 
0466 
0463 
0461 
0458 

10.1284 

1285 
1286 
1287 
1288 

9.8716 
8715 
8714 

8713 
8712 

9-8255 

10.1745 

9-9544 

10.0456 

10.1289 

9.8711 

/ 

m    a 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l 

1" 

Tan. 

Cosec. 

Sin. 

m     a 

/ 

Diff. 

Diflf. 

131°  =  8h  44m  ]                                                                                                    [8h  iam  =  483 

TABLE  IX. 


69 


42° 

=  2h  48m  ]            Log.  8tne$,  Tangents,  and  Secants.           [  9h  8™  --  137° 

!    ni     s 

Sin. 

Diflf. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

HI      S 

/ 

O'.l 

o 

0 
0 

I 

1 

o 

0 

o 
I 

o 

0 

o 
I 
I 
I 

I 
I 

I 

o 

0 

o 

I 

0 

o 
o 

I 

o 

0 

o 

1s 

O'.l 

I8 

i 
i 

2 

' 

2 

' 

I 
2 

I 

2 

I 

2 

1 
I 

2 

0 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
3O 
31 
32 
33 
34 
'  35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
{51 
52 
53 
154 
55 
56 
57 
58 
59 
60 

48    0 
4 
8 
12 
16 
48  2O 
24 
28 
32 
36 
4840 
44 
48 
52 
56 
49    O 
4 
8 
12 
16 
4920 
24 
28 
32 
36 
4940 
44 
48 
52 
56 
5O    O 
4 
8 
12 
16 
50  20 
24 
28 
32 
36 
5O4O 
44 
48 
52 
56 
51    0 
4 
8 
12 
16 
5120 
24 
28 
32 
36 
5140 
44 
48 
52 
56 
52    O 

9-8255 
8257 
8258 
8259 
8261 

o 
o 

0 
0 

I 

o 
I 

o 
I 

10.1745 

1743 
1742 
1741 
1739 

9-9544 
9547 
9549 
9552 
9555 

o 

0 

I 

I 

I 

2 
2 
2 

O 

o 

I 

I 
I 
I 
2 
2 
2 

0 
O 
I 
I 
I 
I 
2 
2 
2 

0 
O 

I 
I 
I 
I 
2 
2 
2 

0 
O 
I 
I 

I 
I 
2 
2 
2 

0 
O 

I 
I 
I 
2 
2 
2 

10.0456 
°453 
045  J 
0448 

0445 

10.1289 
1290 
1292 
1293 
1294 

9.8711 
8710 
8708 

8707 
8706 

12    0 
1156 
52 

48 
1144 
40 
36 
32 
28 
1124 
20 
16 
12 
8 
11     4 
11     0 
1056 
52 
48 
1044 
40 
36 
32 
28 
1024 
20 
16 
12 
8 
10    4 
10    0 
956 
52 
48 
944 
40 
36 
32 
28 
924 
20 
16 
12 
8 
9    4 
9    0 
856 
52 
48 
844 
40 
36 
32 
28 
824 
20 
16 
12 
8 
8    4 
8    0 

60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 

10 

9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

9.8262 
8264 
8265 
8266 
8268 

10.1738 
1736 
1735 
1734 
1732 

9-9557 
9560 
9562 
9565 
_9_567_ 
9-9570 
9572 
9575 
9577 
9580 

10.0443 
0440 
0438 
0435 
0433 

10.1295 
1296 
1297 
1298 
1300 

9.8705 

8704 
8703 
8702 
8700 

9.8269 
8270 
8272 
8273 
8275 

10.1731 
1730 
1728 
1727 
1725 

10.0430 
0428 
0425 
0423 
0420 

10.1301 
1302 
I3°3 
i3°4 
1305 

9.8699 
8698 
8697 
8696 

8695 

9.8276 
8277 
8279 
8280 
8282 

10.1724 

1723 
1721 
1720 
1718 

9.9582 

9585 
9588 
959° 
9593 

10.0418 

0415 
0412 
0410 
0407 

10.1306 
1308 
1309 
1310 
1311 
10.1312 

1313 
I3H 
1316 

I31; 

9.8694 
8692 
8691 

9.8283 
8284 
8286 
8287 
8289 

10.1717 
1716 

1714 

1713 
1711 

9-9595 
9598 
9600 
9603 
9605 

10.0405 
0402 
0400 
0397 
0395 

9.8688 
868; 
8686 
8684 
8683 

9.8290 
8291 
8293 
8294- 
8295 
9.8297 
8298 
8300 
8301 
8302 

10.1710 
1709 
1707 
1706 
1705 

9.9608 
9610 
9613 

9615 
9618 

10.0392 
0390 
0387 
0385 
_03§2_ 

10.0379 
0377 
0374 
0372 
0369 

10.1318 

I3J9 
1320 
1321 

1323 

9.8682 
8681 
8680 
8679 
8677 

10.1703 
1702 
1700 
1699 
1698 

9.9621 
9623 
9626 
9628 
9631 

10.1324 

1325 
1326 
1327 

1328 

9.8676 

8675 
8674 
8673 
8672 

9.8304 
lol 

8308 
8309 

10.1696 

1695 
1694 
1692 
1691 

9-9633 
9636 
9638 
9641 
'   9643 

10.0367 
0364 
0362 
°359 
0357 

10.1329 
i33i 
'332 
'333 
*334 

K^SSS 
1336 
1338 
'339 
*340 

9.8671 
8669 
8668 
8667 
8666 

9.8311 
8312 
8313 
8315 

8316 

10.1689 
1688 
1687 
1685 
1684 

9.9646 
9648 
9651 

9^ 
9656 

10.0354 
0352 
0349 
0347 
0344 

9.8665 
8664 
8662 
8661 
8660 

9-831  7 
8319 
8320 
8322 
8323 

10.1683 
1681 
1680 
1678 
1677 

9.9659 
9661 
9664 
9666 
9669 

10.0341 
0339 
0336 
°334 
0331 

10.1341 

*342 

*343 
'345 
1346 

9.8659 
8658 
8657 

8655 
8654 

9.8324 
8326 

8327 
8328 
8330 

9-8331 
8332 
f334 
8335 
8336 

10.1676 
1674 
1673 
1672 
1670 

9.9671 
9674 
9676 
9679 
9681 

10.0329 
0326 
0324 
0321 
0319 

10.1347 
1348 
1349 
!35o 
J352 

9.8653 
8652 
8651 
8650 
8648 

10.1669 
1668 
1666 
1665 
1664 

9.9684 

9689 
9691 
9694 

10.0316 

°3  J4 
0311 
0309 
0306 

10-1353 
1354 
1355 
1356 
1358 

9.8647 
8646 
8645 
8644 
8642 

9-8338 

10.1662 

9.9697 

10.0303 

Jo.1359 

9.8641 

' 

m    s 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

(X.I 

1s 

Tan. 

Cosec. 

Sin. 

m    s 

i 

Diff. 

Diff. 

132°  =  8h  48m  ]                                                                                                      [  311  gm  _  470  | 

70 


TABLE  IX 


43°  =  2b  52"  ]            Log.  Sines,  Tangents,  and  Secants.           [  9h  4™  =  136° 

i 

m    a 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot. 

Sec. 

Cos. 

m    B 

' 

O'.l 

1" 

o 

I 
I 

o 
I 

o 

I 
I 

o 

I 
I 

o 
I 

I 

0 

I 

I 

O'.l 

1" 

0 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
J28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

52    0 
4 

8 
12 
16 
5220 
24 
28 
32 
36 
5240 
44 
48 
52 
56 
53    0 
4 
8 
12 
16 
5320 
24 
28 
32 
36 
5340 
44 
48 
52 
56 
54    0 
4 
8 
12 
16 
5420 
24 
28 
32 
36 
5440 
44 
48 
52 
56 
55    0 
4 
8 
12 
16 
5520 
24 
28 
32 
36 
5540 
44 
48 
52 
56 
56    O 

9-8338 
8339 
8341 
8342 

8343 

0 

o 

0 
0 

o 
o 
o 
o 

0 

o 

0 

o 

0 

o 
o 

0 

o 
o 
o 

0 
0 

o 
o 

0 

10.1662 
1661 

1659 
1658 

1657 
10.1655 
1654 
1653 
1651 
1650 

9.9697 
9699 
9702 
9704 
9707 

o 
o 
i 

i 
i 
i 

2 
2 

2 

O 
O 
I 
I 
I 
I 
2 
2 
2 

0 
O 

I 
I 
I 
I 
2 
2 
2 

O 
O 
I 
I 
I 
I 
2 
2 
2 

O 
0 
I 
I 
I 
I 
2 
2 
2 

0 
O 

I 

I 
I 
2 
2 
2 

I 
i 

2 

2 

I 
I 

2 

I 
I 

2 

I 
2 

I 
I 
2 

~l-~ 

10.0303 
0301 
0298 
0296 
0293 
10.0291 
0288 
0286 
0283 
0281 

10.1359 
1360 
1361 
1362 
1363 

9.8641 
8640 
8639 
8638 
8637 

8    0 
756 
52 
48 
744 
40 
36 
32 
28 
724 
20 
16 
12 
8 
7    4 
7    0 
656 
52 
48 
644 
40 
36 
32 
28 
624 
20 
16 
12 
8 
6    4 
6    0 
556 
52 
48 
5  21 
40 
36 
32 
28 
524 
20 
16 
12 
8 
5    4 
5    0 
456 
52 
48 
444 
40 
36 
32 
28 
424 
2O 
16 
12 
8 
4    4 
4    O 

60 
59  I 
5N  i 
57 
56 
55 
54 
53 
52 
51 

50 
49 
48  ! 
47 
46 
45 
44| 
43 
42  i 
41 
40 
39 
38 
37  ! 
36  ; 
35 
34 
33 
32 
31 
3O 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18  i 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
0 

9-8345 
8346 

8347 
8349 
8350 

9.9709 
9712 

97U 
9717 

97*9 

10.1365 
1366 
1367 
1368 
1369 

9-8635 
8634 
8633 
8632 
8631 

9.f35i 

8353 
8354 
f355 
8357 

10.1649 
1647 
1646 
1645 
1643 

9.9722 
9724 
9727 
9729 
9732 

10.0278 
0276 
0273 
0271 
0268 

10.1371 
1372 
1373 

1374 
1375 

9.8629 
8628 
8627 
8626 
8625 

9-8358 

8359 
8361 
8362 
8363 

9-8365 
8366 

8367 
8369 
8370 

10.1642 
1641 
1639 
1638 
1637 
10.1635 
J634 
1633 
1631 
1630 

9-9735 
9737 
9740 
9742 
9745 

10.0265 
0263 
0260 
0258 
0255 

10.1376 

1378 
1379 
1380 

1381 
10.1382 
1384 
1385 
1386 

1387 

9.8624 
8622 
8621 
8620 
8619 

9-9747 
975° 
9752 
9755 
9757 

10.0253 
0250 
0248 
0245 
0243 

9.8618 
8616 
8615 
8614 
8613 

9-837I 
8373 
8374 
8375 
8377 

10.1629 
1627 
1626 
1625 
1623 

9.9760 
9762 

9765 
9767 
9770 

10.0240 
0238 

0235 
0233 
0230 

10.1388 
1390 

i39i 
1392 

1393 

9.8612 
8610 

8607 

9-8378 
8379 
8381 
8382 
8383 

10.1622 
1621 
1619 
1618 
1617 

9-9773 
9775 
9778 
9780 

9783 

10.0227 
0225 

O222 
0220 
0217 

10.1394 
1396 
1397 
1398 
1399 

9.8606 
8604 
8603 
8602 
8601 

9-8385 
8386 

8387 
8389 
8390 

10.1615 
1614 
1613 
1611 
1610 

9-9785 
9788 
9790 
9793 
9795 

IO.O2I5 
O2  12 
0210 
0207 
O2O5 

10.1400 
1402 

i4°3 
1404 

1405 

9.8600 
8598 
8597 
8596 
8595 

9-839I 
8393 
8394 
8395 
8397 

10.1609 
1607 
1606 
1605 
1603 

9.9798 
9800 
9803 
9805 
9808 
9.9810 

9813 
9816 
9818 
9821 

IO.O2O2 
O2OO 
0197 

0195 
0192 

10.1406 
1408 
1409 
1410 
1411 

9-8594 
8592 

8591 
8590 
8589 
"9:8588 
8586 
8585 
8584 
8583 

9-8398 

8399 
8401 
8402 
8403 

10.1602 
1601 
1599 
1598 
1597 

10.0190 
0187 
0184 
0182 
0179 

10.1412 
1414 

1415 
1416 
1417 

9.8405 
8406 
8407 
8409 
8410 

10.1595 
1594 
1593 
1591 
!59<> 
10.1589 
1588 
1586 
1585 
1584 

9.9823 
9826 
9828 
9831 
9833 

IO.OI77 
0174 
0172 
0169 
0167 

10.1418 
1420 
1421 
1422 
1423 

9-8582, 
8580 
8579 
8578 
8577 

9.8411 
8412 
8414 

8415 
8416 

9.9836 
9838 
9841 

9843 
9846 

IO.OI64 
Ol62 
0150 
0157 
0154 

10.1425 
1426 
1427 
1428 
1429 

9-8575 
8574 
8573 
8572 

8571 

9.8418 

10.1582 

9.9848 

IO.OI52 

10.1431 

9-8569 

/ 

m     B 

Cos. 

O'.l 

I8 

Sec. 

Cot. 

O'.l 

Tan. 

Cosec. 

Sin. 

m    B 

/ 

Diff. 

Diff. 

133°  =  8h  52m  ]                                                                                                   [  3h  4™  =  46° 

TABLE  IX. 


71 


44°  .--•  2Q  56m  ]            Log.  Sines,  Tangents,  and  Secants.            [  9h  Ora  =  135° 

/ 

m    s 

Sin. 

Diff. 

Cosec. 

Tan. 

Diff. 

Cot 

Sec. 

Cos. 

m     s 

' 

O'.l 

1* 

0.1 

1s 

O 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 

56    0 
4 

8 
12 
16 
5620 
24 
28 
32 
36 
5640 
44 
48 
52 
56 
57    0 
4 
8 
12 
16 
5720 
24 
28 
32 
36 
5740 
44 
48 
52 
56 
58    0 
4 
8 
12 
16 
5820 
24 
28 
32 
36 
58  4O 
44 
48 
52 
56 
59    0 
4 
8 
12 
16 
5920 
24 
28 
32 
36 
5940 
44 
48 
52 
56 
6O    O 

9.8418 
8419 
8420 
8422 
8423 

o 
o 
o 
o 
o 
I 
I 
I 

0 

o 

0 

o 

0 

I 

I 
o 

0 

o 

0 

o 
I 
I 

0 

o 
o 
o 

0 

I 
I 

I 

0 

o 
o 

0 
0 

I 
I 
I 

I 

0 

o 
o 

0 
0 

I 
I 

I 
I 

o 
o 
I 

0 

o 
I 

o 

0 

I 

0 
0 

I 

0 

o 
I  . 

0 

o 
I 

10.1582 
1581 
1580 
1578 
_L577 
10.1576 
1574 
1573 
1572 
i57i 

9.9848 
9851 

9853 
9856 
9858 

0 

o 

2 
2 
2 

O 
O 

I 
I 

I 
2 
2 
2 

O 
0 

I 

I 

2 
2 
2 

O 
O 

I 
I 

2 
2 
2 

O 
O 

I 
I 
I 
I 
2 
2 
2 

0 
O 

I 

I 

2 
2 
2 

i 

2 

I 
I 
2 

I 

2 

I 
I 
2 

I 
I 

2 

I 

2 

10.0152 
0149 
0147 
0144 
0142 

10.1431 
H32 
H33 
H34 
1436 

9.8569 
8568 

8567 
8566 
8564 

9-8563 
8562 
8<6i 
8560 
8558 

4    0 

356 
52 
48 
344 
40 
36 
32 
28 
324 
20 
16 
12 
8 
3    4 
3    0 
256 
52 
48 
244 
40 
36 
32 
28 
224 
20 
16 
12 
8 
2    4 
2    0 
156 
52 
48 
144 
40 
36 
32 
28 
124 
20 
16 
12 
8 
1    4 
1    0 
O56 
52 
48 
044 
4O 
36 
32 
28 
024 
20 
16 
12 
8 
0    4 
0    0 

60  j 
59 

58 

57  ; 

56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 
39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 
9 
8 
7 
6 
5 
4 
3 
2 
1 
O 

9.8424 
8426 
8427 
8428 
8429 

9.9861 
9864 
9866 
9869 
9871 

10.0139 
0136 

0134 
0131 

0129 

10.1437 
1438 
H39 
1440 
1442 

9.84^1 
8432 
f433 
8435 
8436 

10.1569 
1568 
1567 
1565 
1564 
10.1563 
1561 
1560 
1559 
1558 

9.9874 
9876 
9879 
9881 
9884 

10.0126 
0124 

0121 
OII9 

0116 

10.1443 
1444 
1445 
1447 

1448 

9.8557 
8556 

8555 

8553 
8552 

9.8437 
8439 
8440 
8441 
8442 

9.9886 
9889 
9891 
9894 
9896 

10.0114 
oin 
0109 
0106 
0104 

10.1449 
1450 
1452 

J453 
H54 

9-855I 
8550 
8548 
8547 
8546 

9.8444 

8445 
8446 
8448 
8449 

10.1556 
1555 
!554 
1552 
i55i 

9.9899 
9901 
9904 
9907 
9909 

IO.OIOI 

0099 
0096 
0093 
0091 

10.1455 
1456 
H58 
H59 
1460 

9-8545 
8544 
8542 

8541 
8540 

9.8450 
8451 
8453 
8454 
8455 

10.1550 
1549 
1547 
1546 
1545 

9.9912 
9914 
9917 
9919 
9922 

10.0088 
0086 
0083 
0081 
0078 

10.1461 

1463 
1464 

1465 
1466 

9-8539 
8537 
8536 
8535 
8534 

9-8457 
8458 

8459 
8460 
8462 

io-i543 

1542 

i54i 
1540 

1538 

9.9924 
9927 
9929 
9932 
9934 

10.0076 
0073 
0071 
0068 
0066 

10.1468 
1469 
1470 
1471 
H73 

9-8532 
8531 
8530 
8529 
8527 

9.8463 
8464 
8466 
8467 
8468 

io.i537 
1536 
*534 
1533 
1532 

9-9937 
9939 
9942 

9944 
9947 

10.0063 
0061 
0058 
0056 
0053 

10.1474 

1475 
1476 
1478 
H79 

9.8526 

8525 
8524 
8522 
8521 

9.8469 
8471 
8472 
8473 
8475 

10-1531 
1529 

1528 

1527 
1525 

10.1524 

1523 
1522 
1520 
1519 

9-9949 
9952 
9955 
9957 
9960 

10.0051 
0048 
0045 
0043 
0040 

10.1480 
1481 

1483 
1484 

1485 

9-8520 
8519 
8517 
8516 
8515 

9.8476 

8477 
8478 
8480 
8481 

9.9962 

9965 
9967 
9970 
9972 

10.0038 

0035 
0033 
0030 
0028 

10.1486 

1488 

1489 
1490 
1491 

9.8514 
8512 
8511 
8510 
8509 

9.8482 
8483 
8485 
8486 
8487 

10.1518 
1517 
1515 
I5U 
1513 

9-9975 
9977 
9980 
9982 
9985 

10.0025 
0023 

OO2O 
00l8 
0015 

10.1493 
1494 

H95 
1496 
1498 

9.8507 
8506 
8505 
8504 
8502 

9.8489 
8490 
8491 
8492 
8494 

10.1511 
1510 

1509 
1508 
1506 

9.9987 
9990 
9992 
9995 
9997 

IO.OOI3 
OOIO 

0008 

0005 
0003 

10.1499 
1500 
1501 
1503 
1504 

9.8501 
8500 

8499 
8497 
8496 

9.8495 

10.1505 

10.0000 

IO.OOOO 

10.1505 

9.8495 

/ 

m    B 

Cos. 

O'.l 

1s 

Sec. 

Cot. 

O'.l 

1s 

Tan. 

Cosec. 

Sin. 

m    s 

/ 

Diff. 

Diff. 

134°  =  8b  56m  ]                                                                                                   [  3h  Om  -  45° 

72 


TABLE  X. 


JLog.  Sines  to  every  Tenth  of  a  Degree. 

1 

o°.o 

0' 

6 

0°.2 
12' 

0°.3 

18' 

O°.4 
24' 

0°.5 
3O' 

0°.6 

36 

0°.7 
42' 

O°.8 

48' 

0°.9 
54 

60 

egrees. 

5 

Q 

0 

CO 

7.2419 

7.5429 

7.7190 

7-8439 

7.9408 

8.0200 

8.0870 

8.1450 

8.1961 

8.2419 

89 

i 

8.2419 

8.2832 

8.3210 

8.3558 

8.3880 

8.4179 

4459 

4723 

497i 

5206 

5428 

88 

2 

5428 

5640 

5842 

6035 

6220 

6397 

6567 

6731 

6889 

7041 

7188 

87 

3 

7188 

7330 

7468 

7602 

773  i 

7857 

7979 

8098 

8213 

•8326 

8436 

86 

4 

8436  1  8543 

8647 

8749 

8849 

8946 

9042 

9i35 

9226 

9315 

9403 

85 

5 

8.9403 

8.9489 

8-9573 

8.9655 

8.9736 

8.9816 

8.9894 

8.9970 

9.0046 

9.0120 

9.0192 

84 

6 

9.0192 

9.0264 

9-0334 

9.0403 

9.0472 

9-0539 

9.0605 

9.0670 

0734 

0797 

0859 

83 

7 

0859 

0920 

0981 

1040 

1099 

"57 

1214 

1271 

1326 

1381 

1436 

82 

8 

1436 

1489 

1542 

1594 

1646 

1697 

1747 

1797 

1847 

1895 

*943 

81 

9 

1943 

1991   2038 

2085 

2131 

2176 

2221 

2266 

2310 

2353 

2397 

80 

10 
11 

9.2397 
2806 

9.2439 
2845 

9.2482 
2883 

9.2524 
2921 

9-2565 
2959 

9.2606 
2997 

9.2647 
3°34 

9.2687 
3070 

9.2727 
3107 

9.2767 
3*43 

9.2806 

79 

78 

12 
13 
14 

3179 
3521 

3837 

3214 
3554 
3867 

3249   3284 
3586   3618 
3897   3927 

3650 
3957 

3353 
3682 
3986 

3387 
3713 
4015 

3745 
4044 

3455 
3775 
4073 

3488 
3806 
4102 

3837 
413° 

77 

76 
75 

15 
16 
17 

9.4130 
4403 
4659 

9-4I58 

443° 
4684 

9.4186 
4456 
4709 

9.4214 
4482 
4733 

9.4242 
45o8 
4757 

9.4269 

4533 
478i 

9.4296 

4559 
4805 

9-4323 
4584 
4829 

9-4350 
4609 

4853 

9-4377 
4t>34 

4876 

9-4403 
4659 
4900 

74 

73 
72 

,  18 

4900 

4923 

4946   4969 

4992 

5015 

5°37 

5060 

5082 

5104 

5126 

71 

19 

5126 

5H8 

5170 

5192 

5213 

5235 

5256 

5278 

5299 

5320 

5341 

70 

20 

9.5341 

9-536i 

9.5382 

9.5402 

9-5423 

9-5443 

9-5463 

9.5484 

9-5504 

9-5523 

9-5543 

69 

21 
22 

5543 
5736 

5563 
5754 

5583 
5773 

5602 
5792 

5621 
5810 

5641 
5828 

5660 
5847 

5679 
5865 

5883  I  5901 

5736 

68  j 
67 

23 

5919 

5937 

5954 

5972 

5990 

6007 

6024 

6042 

6059 

6076 

6093 

66 

24 

6093 

6110 

6127 

6144 

6161 

6177 

6194 

6210 

6227 

6243 

6259 

65 

25 

9.6259 

9.6276 

9.6292 

9.6308 

9-6324 

9-6340 

9-6356 

9-6371 

9-6387 

9.6403 

9.6418 

64 

26 

6418 

6434 

6449  |  6465 

6480 

6495 

6510 

6526   6541 

6556 

6570 

63  i 

27 

6570 

6585 

6600 

6615 

6629 

6644 

6659 

6673 

6687 

6702 

6716 

62 

28 

6716 

673° 

6744 

6759 

6/73 

6787 

6801 

6814 

6828 

6842 

6856 

61 

29 

6856 

6869 

6883 

6896 

6910 

6923 

6937 

6950 

6963 

6977 

6990 

6O 

30 

9.6990 

9-7003 

9.7016 

9.7029 

9.7042 

9-7055 

9.7068 

9.7080 

9.7093 

9.7106 

9.7118 

59 

31 

7118 

7131 

7144 

7156 

7168 

7181 

7193 

7205 

7218 

7230 

7242 

58 

32 

7242 

7254 

7266 

7278 

7290 

7302 

73  H 

7326 

7338 

7349 

736i 

57 

33 

7361 

7373 

7384 

7396 

7407 

7419 

743° 

7442 

7453 

7464 

7476 

56 

j  34 

7476 

7487 

7498 

75°9 

7520 

7531 

7542 

7553 

7564 

7575 

7586 

55 

!  35 

9-7586 

9-7597 

9.7607 

9.7618 

9.7629 

9.7640 

9-7650 

9.7661 

9.7671 

9.7682 

9.7692 

54 

36 
37 

7692 
7795 

7703 
7805 

7713 
7815 

7723 
7825 

7734 
7835 

7744 
7844 

7754 
7854 

7764 
7864 

7774 
7874. 

7785 
7884 

7795 
7893 

53 

52 

38 

7903   7913   7922 

7932 

794i 

795  i 

7960 

7970 

7979 

7989 

51 

39 

7989 

7998   8007   8017 

8026 

8035 

8044 

8053 

8063 

8072 

8081 

50 

40 
11 

9.8081 
8169 

9.8090 
8178 

9.8099  9.8108  19.8117 
8187   8191;  !  8204 

9.8125 
8213 

9.8134 
8221 

9.8143 
8230 

9.8152 
8238 

9.8161 
8247 

9.8169 

8255 

49 

48 

42 
43 

8338 

8264 
8346 

8272 
8354 

8280   8289 
8362   8370 

8297 
8378 

8305 
8386 

8313 
8394 

8322 
8402 

8330 
8410 

S$ 

47 
46 

44 

8418 

8426 

8433 

8441   8449 

8457 

8464 

8472 

8480 

8487 

8495 

45 

03 

c/i 

60    54' 

48' 

42'   36 

30 

24' 

18' 

12' 

6 

0 

<D 

E 

t 

8 

0) 

1  .0 

0°.9 

OD.8 

0°.7 

0.6 

0.5 

0.4 

0°.3 

0°.2 

O°.l 

o°.o 

w 

p 

Q 

Logarithmic  Cosines. 

TABLE  X. 


73 


Log.  Sines  to  every  Tenth  of  a  Degree. 

02 

o°.o 

0°1 

0°.2 

0°.3 

0°.4 

0°.5 

0.6 

0°.7 

0°.8 

0°.9 

I'O 

«  1 
I 

1 

O' 

6 

12' 

18' 

24' 

3O' 

36' 

42' 

48< 

54' 

60' 

H 

m 

o 
Q 

i  45 

9.8495 

9.8502 

9.8510 

9-8517 

9-8525 

9-8532 

9.8540 

9-8547 

9-8555 

9.8562 

9.8569 

44 

46 

8569 

8577 

8584 

8591 

8598 

8606 

8613 

8620 

8627 

8634 

8641 

43 

8641 

8648   8655 

8662 

8669 

8676 

8683 

8690 

8697 

8704 

8711 

42 

1  48 

8711  j 

8718   8724 

8731 

8738 

8745 

8751 

8758 

8765 

8771 

8778 

41 

49 

8778 

8784* 

8791 

8797 

8804 

8810 

8817 

8823 

8830 

8836 

8843 

40 

50 
51 

9.8843 
8905 

9.8849 
8911 

9-8855 
8917 

9.8862 
8923 

9.8868 
8929 

9.8874 
8935 

9.8880 
8941 

9.8887 
8947 

9.8893 
8953 

9.8899 
8959 

9.8905 
8965 

39 

38 

1  52 

8965 

8971 

8977 

8983 

8989 

8995 

9000 

9006 

9012 

9018 

9023 

37 

53 

9023 

9029 

9°35 

9041 

9046 

9052 

9°57 

9063 

9069 

9074 

9080 

36 

54 

9080 

9085 

9091 

9096 

9101 

9107 

9112 

9118 

9123 

9128 

9134 

35 

55 

9-9I34 

9-9I39 

9.9144 

9.9149 

9.9I55 

9.9160 

9.9165 

9.9170 

9-9I75 

9.9181 

9.9186 

34 

56 

9186 

9191 

9196 

9201 

9206 

9211 

9216 

9221 

9226 

9231 

9236 

33 

!  57 

9236 

9241 

9246 

9251 

9255 

9260 

9265 

9270 

9275 

9279 

9284 

32  ! 

i  58 

9284 

9289  |  9294 

9298 

93°3 

9308 

9312 

93J7 

9322 

9326 

933  i 

31 

59 

933  i 

9335   934° 

9344 

9349 

9353 

9358 

9362 

9367 

937i 

9375 

30 

60 

9-9375 

9.9380 

9-9384 

9.9388 

9-9393 

9-9397 

9.9401 

9.9406 

9.9410 

9.9414 

9.9418 

29 

61 

9418 

9422   9427 

943  i 

9435 

9439 

9443 

9447 

9451 

9455 

9459 

28 

62 

9459 

9463   9467 

9471 

9475 

9479 

9483 

9487 

9491 

9495 

9499 

27 

63 

9499 

9503   9506 

9510 

9SJ4 

9518 

9522 

9525 

9529 

9533 

9537 

26 

64 

9537 

954° 

9544 

9548 

955  i 

9555 

9558 

9562 

9566 

9569 

9573 

25 

65 

9-9573 

9-9576 

9.9580 

9-9583 

9.9587 

9.9590 

9-9594 

9-9597 

9.9601 

9.9604 

9.9607 

24 

66 

9607 

9611 

9614 

9617 

9621 

9624 

9627 

9631 

9634 

9637 

9640 

23 

67 

:  68 

9640 
9672 

9643 
9675 

9647 
9678 

9650 
9681 

9653 
9684 

$8 

9659 
9690 

9662 
9693 

9696 

9669 
9699 

9672 
9702 

22 
21 

69 

9702 

9704 

9707 

9710 

9713 

9716 

9719 

9722 

9724 

9727 

973° 

20 

7O 

9-973° 

9-9733 

9-9735 

9-9738 

9.9741 

9-9743 

9.9746 

9-9749 

9-9751 

9-9754 

9-9757 

19 

71 

9757 

9759 

9762 

9764 

9767 

9770 

9772 

9775 

9777 

9780 

9782 

18 

72 

9782 

9785 

9787 

9789 

9792 

9794 

9797 

9799 

9801 

9804 

9806 

17 

73 

9806 

9808 

9811 

9815 

9817 

9820 

9822 

9824 

9826 

9828 

16 

i  74 

9828 

9831 

9833 

9835 

9837 

9839 

9841 

9843 

9845 

9847 

9849 

15 

75 

9-9849 

9.9851 

9-9853 

9.9855 

9.9857 

9.9859 

9.9861 

9.9863 

9.9865 

9.9867 

9.9869 

14 

76 

9869 

9871 

9873 

9875 

9876 

9878 

9880 

9882 

9884 

9885 

9887 

13 

77 

9887 

9889 

9891 

9892 

9894 

9896 

9897 

9899 

9901 

9902 

9904 

12 

1  T8 

9904 

9906 

9907 

9909 

9910 

9912 

9913 

9915 

9916 

9918 

9919 

11 

1  79 

•9919 

9921 

9922 

9924 

9925 

9927 

9928 

9929 

993  i 

9932 

9934 

1O 

!  80 

9-9934 

9-9935 

9.9936 

9-9937 

9-9939 

9.9940 

9.9941 

9-9943 

9-9944 

9-9945 

9.9946 

9 

81 

9946 

9947 

9949 

995° 

995  * 

9952 

9953 

9954 

9955 

9956 

9958 

8 

:  82 

9958 

9959 

9960 

9961 

9962 

9963 

9964 

9965 

9966 

9967 

9968 

7 

83 

9968 

9968 

9969 

9970 

9971 

9972 

9973 

9974 

9975 

9975 

9976 

6 

'  84 

9976 

9977 

9978 

9978 

9979 

9980 

998i 

9982 

9983 

9983 

5 

;  85 

9.9983 

9.9984 

9.9985 

9.9985 

9.9986 

9.9987 

9.9987 

9.9988 

9.9988 

9-9989 

9-9989 

4 

86 

9989 

999° 

999° 

9991 

9991 

9992 

9992 

9993 

9993 

9994 

9994 

3 

87 

9994 

9994 

9995 

9995 

9996 

9996 

9996 

9996 

9997 

9997 

9997 

2 

88 

9997 

9998 

9998 

9998 

9998 

9999 

9999 

9999 

9999 

9999 

9999 

1 

89 

9999 

9999 

IO.OOOO 

IO.OOOO 

IO.OOOO 

10.0000 

IO.OOOO 

10.0000 

IO.OOOO 

10.0000 

IO.OOOO 

O 

;  i 

60 

54' 

48' 

42' 

36' 

30' 

24 

IS' 

12' 

6' 

0' 

o! 

1 

\  I 

1°0 

0°.9 

08 

0°.7 

0°.6 

0°.5 

0°.4 

0°.3 

0°.2 

01 

ooo 

I 

Logarithmic  Cosines. 

74 


TABLE  XL 


Log.  Secants  to  every  Tenth  of  a  Degree. 

t 

I 
% 

o°.o 

0' 

0.1 

6' 

0°.2 
12' 

0.3 

18' 

0°.4 
24' 

0°.5 
30 

0.6 
36 

O°.7 
42 

0^.8 
48' 

O°.9 
54' 

60' 

1 

Q 

Q 

0 

lo.opoo 

IO.OOOO 

10.0000 

IO.OOOO 

IO.OOOO 

IO.OOOO 

10.0000 

10.0000 

IO.OOOO 

IO.OOOI 

IO.OOOI 

89 

1 

OOOI 

OOOI 

OOOI 

OOOI 

OOOI 

OOOI 

0002 

OOO2 

0002 

OOO2 

0003 

88 

2 

0003 

0003 

0003 

0003 

0004 

0004 

0004 

0005 

0005 

0006 

0006 

87 

3 

0006 

0006 

0007 

0007 

0008 

0008 

0009 

0009 

OOIO 

OOIO 

OOII 

86 

4 

OOII 

OOII 

OOI2 

0012 

0013 

0013 

0014 

0015 

0015 

0016 

0017 

85 

5 

10.0017 

10.0017 

10.0018 

10.0019 

10.0019 

IO.OO2O 

IO.OO2I 

IO.OO22 

IO.OO22 

10.0023 

10.0024 

84 

6 

0024 

0025 

0025 

0026 

0027 

0028 

0029 

0030 

0031 

0032 

0032 

83 

7 

0032 

0033 

0034 

0035 

0036 

0037 

0038 

0039 

OO4O 

0041 

0042 

82 

8 

0042 

0044 

0045 

0046 

0047 

0048 

0049 

0050 

0051 

°°53 

0054 

81 

9 

0054 

0055 

0056 

0057 

0059 

0060 

OO6l 

0063 

OOO4 

0065 

0066 

80 

10 

10.0066 

10.0068 

10.0069 

10.0071 

10.0072 

10.0073 

IO.OO75 

10.0076 

10.0078 

10.0079 

10.0081 

79 

11 

0081 

0082 

0084 

0085 

0087 

0088 

0090 

0091 

0093 

0094 

0096 

78 

12 

0096 

0098 

0099 

OIOI 

0103 

0104 

OIO6 

0108 

OI09 

on  i 

0113 

77 

13 

0113 

0115 

0116 

0118 

OI2O 

OI22 

OI24 

OI25 

0127 

0129 

0131 

76 

14 

0131 

0133 

0135 

0137 

0139 

0141 

OH3 

0145 

0147 

0149 

0151 

75 

15 
16 

10.0151 
0172 

10.0153 

0174 

10.0155 
0176 

10.0157 
0178 

10.0159 

0180 

10.0161 
0183 

10.0163 
0185 

IO.OI65 
Ol87 

IO.OI67 
0189 

10.0169 
0192 

10.0172 

0194 

74 

73 

17 

0194 

0196 

0199 

0201 

0203 

0206 

0208 

O2II 

0213 

0215 

0218 

72 

18 

0218 

O22O 

0223 

0225' 

0228 

0230 

0233 

0236 

0238 

0241 

0243 

71 

19 

0243 

0246 

0249 

0251 

0254 

0257 

0259 

O262 

0265 

0267 

0270 

70 

20 

10.0270 

10.0273 

10.0276 

10.0278 

10.0281 

10.0284 

IO.O287 

IO.O29O 

10.0293 

10.0296 

10.0298 

69 

21 

0298 

0301 

0304 

0307 

0310 

°3*3 

0316 

0319 

0322 

0325 

0328 

68 

22 

0328 

0331 

0334 

0338 

0341 

0344 

°347 

0350 

°353 

°357 

0360 

67 

23 

0360 

0303 

0366 

0369 

°373 

0376 

0379 

0383 

0386 

0389 

°393 

66 

24 

0393 

0396 

0399 

0403 

0406 

0410 

0413 

0417 

0420 

0424 

0427 

65 

25 

10.0427 

10.0431 

10.0434 

10.0438 

10.0442 

10.0445 

10.0449 

10.0452 

10.0456 

10.0460 

10.0463 

64 

26 

0463 

0467 

0471 

0475 

0478 

0482 

0486 

0490 

0494 

0497 

0501 

63 

27 

0501 

0505 

0509 

0513 

0517 

0521 

°525 

0529 

°533 

°537 

0541 

62 

28 

0541 

0545 

0549 

°553 

°557 

0561 

0565 

0569 

°573 

0578 

0582 

61 

29 

0582 

0586 

0590 

°594 

°599 

0603 

0607 

0612 

0616 

0620 

0625 

6O 

30 

10.0625 

10.0629 

10.0633 

10.0638 

10.0642 

10.0647 

10.0651 

10.0656 

10.0660 

10.0665 

10.0669 

59 

31 

0669 

0674 

0678 

0683 

0688 

0692 

0697 

0702 

0706 

0711 

0716 

58 

32 

0716 

0721 

0725 

0730 

0735 

0740 

°745 

0749 

°754 

°759 

0764 

57 

33 

0764 

0769 

0774 

0779 

0784 

0789 

0794 

0799 

0804 

0809 

0814 

56 

34 

0814 

0819 

0825 

0830 

0835 

0840 

0845 

0851 

0856 

0861 

0866 

55 

35 

10.0866  i  1  0.08  72 

10.0877 

10.0882 

10.0888 

10.0893 

10.0899 

10.0904 

10.0909 

10.0915 

10.0920 

54 

36 
37 

0920 
0977 

0926 
0982 

$8 

0937 
0994 

0943 

IOOO 

0948 
1005 

°954 
ion 

0959 
IO17 

,0965 
1023 

0971 
1029 

0977 
I035 

53 

52  • 

38 

1035 

I04I 

1047 

1053 

1059 

1065 

1071 

1077 

1083 

1089 

1095 

51  ! 

39 

1095 

IIOI 

1107 

1113 

1  120 

1126 

1132 

1138 

H45 

"5* 

1157 

50  | 

40 

10.1157 

10.1164 

10.1170 

10.1177 

10.1183 

10.1190 

10.1196 

IO.I2O3 

10.1209 

10.1216 

IO.I222 

49 

41 

1222 

1229 

I235 

1242 

1249 

1255 

1262 

1269 

1276 

1282 

'   1289 

48 

42 

1289 

1296 

'3°3 

1310 

13*7 

1331 

1338 

J345 

'352 

*359 

47 

43 

1359 

1366 

1373 

1380 

1387 

1394 

1402 

1409 

1416 

1423 

143  I 

46  j 

44 

1431 

1438 

1445 

1453 

1460 

1468 

1475 

H83 

1490 

1498 

1505 

45 

a 

u  i 

1 

60' 

54' 

48' 

42' 

36 

30' 

24 

18' 

12 

6 

0 

8  : 

I 

10 

0°.9 

0°.8 

0°.7 

0°.6 

0°.5 

0°.4 

O°.3 

0°.2 

' 

O°.O 

| 

o  i 

Logarithmic  Cosecants. 

TABLE  XL 


75 


Log.  Secants  to  every  Tenth  of  a  Degree. 

|                                               . 

egreea. 

Oa.O 
0 

6 

0°.2 
12' 

0°.3 

18' 

0°.4 
24 

0°.5 
30 

0°.6 
36' 

0°.7 
42 

0°.8 
48' 

0°.9 
54' 

1  .O 

60 

II 

Q 

Q 

!  45 
46 

10.1505 

1582 

10.1513 
1590 

10.1520 
1598 

10.1528 
1606 

10.1536 
1614 

1  o.i  543 

1622 

10.1551 
1630 

10.1559 
1638 

10.1567 
1646 

10.1574 
1654 

10.1582 
1662 

44 
43 

47 

1662 

1670 

1678 

1687 

1695 

1703 

1711 

1720 

1728 

1745 

42 

48 

1745 

1753 

1762 

1770 

1779 

1787 

1796 

1805 

1813 

1822 

1831 

41 

49 

1831 

1839 

1848 

1857 

1866 

1875 

1883 

1892 

1901 

1910 

1919 

40 

50 

10.1919 

10.1928 

10.1937 

10.1947 

10.1956 

10.1965 

10.1974 

10.1983 

10.1993 

IO.2OO2 

IO.2OII 

39 

i  51 

201  1 

202  1 

2030 

2040 

2049 

2058 

2068 

2078 

2087 

2097 

2107 

38 

52 

2IO7 

2116 

2126 

2136 

2146 

2156 

2165 

2175 

2185 

2195 

22O5 

37 

53 

2205 

2215 

2226 

2236 

2246 

2256 

2266 

2277 

-  2287 

2297 

2308 

36 

54 

2308 

2318 

2329 

2339 

2350 

2360 

2371 

2382 

2393 

2403 

2414 

35 

55 

10.2414 

10.2425 

10.2436 

10.2447 

10.2458 

10.2469 

10.2480 

10.2491 

10.2502 

10.2513 

10.2524 

34 

56 

2524 

2536 

2547 

2558 

2570 

2581 

2593 

2604 

2616 

2627 

2639 

33 

57 

2639 

2651 

2662 

2674 

2686 

2698 

2710 

2722 

2734 

2746 

2758 

32 

58 

2758 

2770 

2782 

2795 

2807 

2819 

2832 

2844 

2856 

2869 

2882 

31 

59 

2882 

.  2894 

2907 

2920 

2932 

2945 

2958 

2971 

2984 

2997 

3010 

30 

60 

IO.3OIO 

10.3023 

10.3037 

10.3050 

10.3063 

10.3077 

10.3090 

10.3104 

10.3117 

10.3131 

10.3144 

29 

1  61 

3144 

3158 

3172 

3186 

3199 

3213 

3227 

3241 

3256 

3270 

3284 

28 

;  62 

3284 

3298 

3313 

3327 

334i 

3356 

3385 

3400 

3415 

343° 

27 

1  63 

3430 

3444 

3460 

3474 

349° 

3505 

3520 

3535 

355* 

3566 

3582 

26 

64 

3582 

3597 

36i3 

3629 

3644 

3660 

3676 

3692 

37o8 

3724 

3741 

25 

65 

10.3741 

10-3757 

10.3773 

10.3790 

10.3806 

10.3823 

10.3839 

10.3856 

10.3873 

10.3890 

10.3907 

24 

66 

3907 

3924 

394  l 

3958 

3976 

3993 

4010 

4028 

4046 

4063 

4081 

23 

67 

4081 

4099 

4117 

4135 

4153 

4172 

4190 

4208 

4227 

4246 

4264 

22 

68 

4264 

4283 

4302 

4321 

4340 

4359 

4379 

4398 

4437 

4457 

21 

69 

4457 

4477 

4496 

45  l6 

4537 

4557 

4577 

4598 

4618 

4639 

4659 

20 

70 

10.4659 

10.4680 

10.4701 

10.4722 

10.4744 

10.4765 

10.4787  '10.4808 

10.4830 

10.4852 

10.4874 

19 

71 

4874 

4896 

4918 

4940 

4963 

4985 

5008 

5031 

5054 

5077 

5100 

18 

72 

5100 

5124 

5H7 

5171 

5195 

5219 

5243 

5267 

5291 

534* 

17 

73 

534i 

5366 

5391 

5416 

5467 

5492 

5544 

5570 

5597 

16 

74 

5597 

5623 

5677 

5704 

5758 

5786 

58H 

5842 

5870 

15 

75 
76 

10.5870 
6163 

10.5898 
6194 

10.5927 
6225 

10.5956 
'6255 

10.5985 
6287 

10.6014 
6318 

10.6043 
635° 

10.6073 
6382 

10.6103 
6414 

10.6133 
6446 

10.6163 
6479 

14 
13 

77 

6479 

6512 

6545 

6579 

6613 

6647 

6681 

6716 

6750 

6786 

6821 

12  I 

78 

6821 

6857 

6893 

6930 

6966 

7003 

7041 

7079 

7117 

7155 

7194 

11 

79 

7i94 

7233 

7273 

73  J3 

7353 

7394 

7435 

7476 

75i8 

7561   7603 

10 

80 

10.7603 

10.7647 

10.7690 

10.7734 

10.7779 

10.7824 

10.7869 

10.7915 

10.7962 

10.8009 

10.8057 

9 

81 

82 

8057 
8564 

8105 
8619 

8674 

8203 
8729 

8253 
8786 

8843 

8354 
8901 

8406 
8960 

8458 
9019 

8511 
9080 

8564 
9141 

8 

7 

83 

9141 

9203 

9266 

933° 

9395 

9461 

9528 

9597 

9666 

9736 

9808 

6 

84 

9808 

9880 

9954 

11.0030 

11.0106 

11.0184 

11.0264 

H.0345 

11.0427 

11.0511 

11.0597 

5 

85 
86 

11.0597 
1564 

11.0685 
1674 

11.0774 

1787 

11.0865 
1902 

11.0958 

2021 

11.1054 
2143 

11.1151 

2269 

11.1251 

2398 

"•1353 

2532 

II.H57 
2670 

11.1564 
2812 

4 

3 

87 

2812 

2959 

3111 

3269 

3433 

3780 

3965 

4158 

4360 

4572 

2 

88 

4572 

4794 

5029 

5277 

5541 

5821 

6120   6442 

6790 

7168   7581 

1 

89 

758i 

8039 

8550 

9130 

9800 

12.0592 

12.1561 

12.2810 

12.4571 

12.7581 

-f  :o 

0 

to 
p 

60' 

54' 

48' 

42 

36 

30 

24'   18' 

12 

6 

0 

i 

Q 

10 

0°.9 

0°.8 

0°.7 

0D.6 

0°5 

0°.4   0°.3 

O°.2 

01 

0  0 

1 

Logarithmic  Cosecants. 

70 


TABLE  XII. 


Logarithmic  Tangents  to  every  Tenth  of  a  Degree. 

t 

i 

00 

0°.l 

0°.2 

0°.3 

0.4 

0.5 

0°.6 

03.7 

03.8 

0°.9 

10 

i 

00 
0> 

0 

6 

12' 

18' 

24' 

30 

36 

42 

48' 

54' 

60' 

Q 

0 

1 

—  oo 
8.2419 

7.2419 
8-2833 

7.5429 
8.3211 

7.7190 

7.8439 
8.3881 

7.9409 
8.4181 

8.O2OO 

4461 

8.0870 
4/25 

8.1450 
4973 

8.1962 
5208 

8.2419 

89 

88 

2 

543  l 

5643 

5845 

6038 

6223 

6401 

6571 

6736 

6894 

7046 

7*94 

87 

3 
4 

7  IQ4 
8446 

7337 
8554 

7609 
8762 

7739 
8862 

7865 
8960 

7988 

9056 

8107 
915° 

8223 
9241 

8336 

8446 
9420 

86 

85 

5 

8.9420 

8.9506 

8.9591 

8.9674 

8.9756 

8.9836 

8.9915 

8.9992 

9.0068 

9.0143 

9.0216 

84 

6 

9.0216 

9.0289 

9.0360 

9.0430 

9.0499 

9-0507 

9.0633 

9.0699 

0764 

0828 

0891 

83 

7 

0891 

0954 

1015 

1076 

"35 

1194 

1252 

1310 

1367 

1423 

1478 

82 

8 

1478 

'533 

1587 

1640 

1693 

1745 

1797 

1848 

1898 

1948 

1997 

81 

9 

1997 

2040 

2094 

2142 

2189 

2230 

2282 

2328 

2374 

2419 

2463 

80 

10 

9.2463 

9.2507 

9-255I 

9.2594 

9.2637 

9.2680 

9.2722 

9-2764 

9.2805 

9.2846 

9.2887 

79 

11 

2887 

2927 

2967 

3006 

3040 

3085 

3123 

3162 

3200 

3237 

3275 

78 

12 

3275 

3312 

3349 

3385 

3422 

3458 

3493 

3529 

3564 

3599 

3634 

77 

13 

3634 

3668 

3702 

3736 

3770 

3804 

3837 

3870 

3903 

3935 

3968 

76 

14 

3968 

4000 

4032 

4064 

4095 

4127 

4158 

4189 

4220 

4250 

4281 

75 

15 

9.4281 

9-43  ii 

9-4341 

9-4371 

9.4400 

9.4430 

9-4459 

9.4488 

9-45  1  7 

9.4546 

9-4575 

74 

16 

4575 

4603 

4632 

4660 

4688 

4716 

4744 

4771 

4799 

4826 

4853 

73 

17 

4853 

4880 

4907 

4934 

4961 

4987 

5OI4 

5040 

5066 

5092 

5118 

72 

18 

5118 

5H3 

5l69 

5195 

5220 

5245 

5270 

5295 

5320 

5345 

5370 

71 

19 

5370 

5394 

5419 

5443 

5467 

55i6 

5539 

5563 

5587 

5611 

70 

20 

9.5611 
5842 

9-5634 
5864 

9-5658 

5887 

9-5681 
5909 

9.5704 
5932 

5954 

9-5750 
5976 

9-5773 
5998 

9.5796 
6020 

9.5819 
6042 

9.5842 
6064 

69 

68 

22 

6064 

6086   6108 

6129 

6151 

6172 

6194 

6215 

6236 

6257 

6279 

67 

23 

6279 

6300 

6321 

6341 

6362 

6383 

6404 

6424 

6445 

6465 

6486 

66 

24 

6480 

6506 

6527 

6547 

6567 

6587 

6607 

6027 

6647 

6667 

6687 

65 

25 

9.6687 

9.6706 

9.6726 

9.6746 

9.6765 

9.6785 

9.6804 

9.6824 

9.6843 

9.6863 

9.6882 

64 

26 

6882 

6901 

6920 

6939 

6958 

6977 

6996 

7015 

7034 

7053 

7072 

63 

27 

7072 

7090 

7109 

7128 

7146 

7165 

7183 

7202 

7220 

7238 

7257 

62 

28 

7257 

7275 

7293 

73" 

733° 

7348 

7366 

7384 

7402 

7420 

7438 

61 

29 

7438 

7455 

7473 

7491 

7509 

7526 

7544 

7562 

7579 

7597 

7614 

6O 

30 

9.7614 

9.7632 

9.7649 

9.7667 

9.7684 

9.7701 

9.7719 

9-7736 

9-7753 

9.77-1 

9.7788 

59 

31 

7788 

7805 

7822 

7856 

7873 

7890 

7907 

7924 

7958 

58 

32 
33 

7958 
8125 

7975 
8142 

7992 
8158 

8i75 

8025 
8191 

8042 
8208 

8059 
8224 

8241 

8092 
8257 

8109 
8274 

8125 
8290 

57 
56 

34 

8290 

8306 

8323 

8339 

8355 

8371 

8388 

8404 

8420 

8436 

8452 

55 

35 
36 

9.8452 
8613 

9.8468 
8629 

9.8484 
8644 

9.8501 
8660 

9-8517 
8676 

9.8533 
8092 

9.8549 
8708 

9.8565 
8724 

9.8581 
8740 

9.8597 
8755 

9.8613 
8771 

54 

53 

37 

8771 

8787 

8803 

8818 

8834 

8850 

8865 

8881 

8897 

8912 

8928 

52 

31 

8928 

8944 

8959 

8975 

8990 

9006 

9022 

9037 

9053 

9068 

9084 

51 

39 

9084 

9099 

9"5 

9130 

9146 

9161 

9176 

9192 

9207 

9223 

9238 

5O 

40 

9.9238 

9.9254 

9.9269 

9.9284 

9.9300 

9-93  1  5 

9-933° 

9.9346 

9.9361 

9.9376 

9.9392 

49 

41 

9392 

9407 

9422 

9438 

9453 

9468 

9483 

9499 

95  14 

9529 

9544 

48 

42 

9544 

956o 

9575 

9590 

9605 

9621 

9636 

9651 

9666 

9681 

9697 

47 

43 

44 

9697 
9848 

9712 
9864 

9727 
9879 

9742 
9894 

9757 
9909 

9772 
9924 

9788 
9939 

9803 
9955 

9818 
9970 

9833 
9985 

9848 

10.0000 

46 
45 

03 

03 

S 

60 

54' 

48' 

42' 

36' 

30' 

24' 

18' 

12' 

6' 

O' 

! 

1  .0 

0°.9 

0°.8 

0°.7 

0°.6 

05 

04 

0°.3 

0°.2 

01 

o°.o 

00 

IS 

Logarithmic  Cotangents. 

TABLE  XII. 


<  < 


Logarithmic  Tangents  to  ei'ery  Teuth  of  a  Degree. 

Degrees. 

o°.o 

O' 

0°.l 
6' 

0°.2 
12' 

0°.3 
IN 

0°.4 
24' 

0D.5 
3O' 

0°.6 
36 

0°.7 
42' 

0°.8 
48' 

0°.9 
54 

1°.O 
6O' 

Degrees,  j 

45 

IO.OOOO 

10.0015 

10.0030 

10.0045 

1  0.006  1 

10.0076 

10.0091 

10.0106 

IO.OI2I 

10.0136 

10.0152 

44 

46 

0152 

0167 

0182 

0197 

O2I2 

0227 

0243 

0258 

0273 

0288 

0303 

43 

4? 

0303 

0319 

0334 

0349 

0364 

0379 

0395 

0410 

0425 

0440 

0456 

42 

4S 

0456 

0471 

0486 

0501 

0517 

0532 

0547 

0562 

0578 

0593 

0608 

41 

49 

0608 

0624 

0639 

0654 

0670 

0685 

0700 

0716 

0731 

0746 

0762 

40 

50 

10.0762 

10.0777 

10.0793 

10.0808 

10.0824 

10.0839 

10.0854 

10.0870 

10.0885 

10.0901 

10.0916 

39 

51 

0916 

0932 

0947 

0963 

0978 

0994 

IOIO 

1025 

IO4I 

1056 

1072 

38 

52 

1072 

1088 

1103 

1119 

"35 

1150 

1166 

1182 

"97 

1213 

1229 

37 

53 

1229 

1245 

1260 

.  1276 

1292 

1308 

1324 

1340 

1356 

i37i 

1387 

36 

54 

1387 

1403 

1419 

H35 

H5  i 

1467 

1483 

1499 

1516 

1532 

1548 

35 

55 

10.1548 

10.1564 

10.1580 

10.1596 

10.1612 

10.1629 

10.1645 

10.1661 

10.1677 

10.1694 

10.1710 

34 

56 

1710 

1726 

1743 

1759 

1776 

1792 

1809 

1825 

1842 

1858 

1875 

33 

57 

1875 

1891 

1908 

1925 

1941 

1958 

1975 

1992. 

2008 

2025 

2042 

32 

58 

2042 

2059 

2076 

2093 

2IIO 

2127 

2144 

2161 

2178 

2195 

2212 

31 

59 

2212 

2229 

2247 

2264 

228l 

2299 

2316 

2333 

235  J 

2368 

2386 

30 

60 

10.2386 

10.2403 

10.2421 

10.2438 

10.2456 

10.2474 

10.2491 

10.2509 

10.2527 

10.2545 

10.2562 

29 

61 

2562 

2580 

2598 

2616 

2634 

2652 

2670 

2689 

2707 

2725 

2  743 

28 

62 

2743 

2762 

2780 

2798 

2817 

2835 

2854 

2872 

2891 

2910 

2928 

27 

63 

2928 

2947 

2966 

2985 

3004 

3023 

3042 

3601 

3080 

3°99 

3118 

26 

64 

3Il8 

3*37 

3i57 

3176 

3196 

3215 

3235 

3254 

3274 

3294 

3313 

25 

65 

10.3313 

10-3333 

IO-3353 

10-3373 

10-3393 

10.3413 

10-3433 

io-3453 

10.3473 

10.3494 

10.3514 

24 

66 

35H 

3535 

3555 

3576 

3596 

3617 

3638 

3659 

3679 

3700 

3721 

23 

67 

3721 

3743 

3764 

3785 

3806 

3828 

3849 

3871 

3892 

39J4 

3936 

22 

68 

3936 

3958 

398o 

4002 

4024 

4046 

4068 

4091 

4"3 

4136 

4158 

21 

69 

4I58 

4181 

4204 

4227 

4250 

4273 

4296 

43J9 

4342 

4366 

4389 

20 

7O 

10.4389 

10.4413 

10.4437 

10.4461 

10.4484 

10.4509 

10-4533 

10-4557 

10.4581 

10.4606 

10.4630 

19 

71 

4630 

4655 

4680 

4705 

473° 

4755 

478o 

4805 

4831 

4857 

4882 

18 

72 

4882 

4908 

4934 

4960 

4986 

5013 

5°39 

5066 

5093 

5120 

5H7 

17 

73 

5'47 

5174 

5201 

5229 

5256 

5284 

5312 

5340 

5368 

5397 

5425 

16 

74 

5425 

5454 

5483 

5512 

5541 

557o 

5600 

5629 

5659 

5689 

5719 

15 

75 

10.5719 

10-5750 

10.5780 

10.5811 

10.5842 

10.5873 

10.5905 

10.5936 

10.5968 

10.6000 

10.6032 

14 

76 

6032 

6065 

6097 

6130 

6163 

6196 

6230 

6264 

6298 

6332 

6366 

13 

|  77 

6366 

6401 

6436 

6471 

6507 

6542 

6578 

6615 

6651 

6688 

6725 

12 

78 

6725 

6763 

6800 

6838 

6877 

6915 

6954 

6994 

7033 

7073 

7"3 

11 

79 

7H3 

7i54 

7195 

7236 

7278 

7320 

7363 

7406 

7449 

7493 

7537 

1.0 

80 

io.7537 

10.7581 

10.7626 

10.7672 

10.7718 

10.7764 

10.7811 

10.7858 

10.7906 

10.7954 

10.8003 

9  I 

81 

8003 

8052 

8102 

8152 

8203 

8255 

8307 

8360 

8413 

8467 

8522 

8  ! 

82 

8522 

8577 

8633 

8690 

8748 

8806 

8865 

8924 

8985 

9046 

9109 

7 

83 

9109 

9172 

9236 

9301 

9367 

9433 

95°  i 

957° 

9640 

9711 

9784 

6 

84 

9784 

9857 

9932 

11.0008 

1  1  .0085 

11.0164 

11.0244 

11.0326  11.0409 

11.0494 

11.0580 

5 

85 

11.0580 

11.0669 

11.0759 

11.0850 

11.0944 

11.1040 

11.1138 

11.1238111.1341 

11.1446 

"•1554 

4 

86 

1554 

1664 

1777 

1893 

2OI2 

2135 

2261 

2391  !  2525 

2663 

2806 

3 

87 

2806 

2954 

3106 

3264 

3429 

3599 

3777 

3962  j  4155 

4357 

4569 

2 

1  88 

4569 

4792 

5027 

5275 

5539 

5819 

6119 

6441  i  6789 

11.7167 

758i 

1 

I  89 

7581 

8038 

8550 

9130 

9800 

12.0591 

12.1561 

12.2810 

12.4571 

12.7581 

-f-  CO 

0 

i 

6O' 

54' 

48' 

42' 

36 

30 

24 

18' 

12' 

6 

O' 

02 

$ 

& 

0) 

1°.0   0°.9 

0°.8 

0°.7 

0°.6 

0°.5 

O°.4   O°.3 

0°.2 

0°.l 

o°.o 

I 

V 

Q 

P 

Logarithmic  Cotangeuts. 

78 


TABLE  XIII. 


Log.  Tangents  and  Cotangents  in  Time. 

ou 

Diff. 

lb 

Diff. 

2h 

Diff. 

3h 

Diff. 

4h 

Diff. 

5h 

Diff. 

m 

6h 

Om.l 

l^h 

Ora.l 

8h 

Om.l 

9h 

Ora.l 

10 

Om.l 

llh 

6m.i 

m 

0 

—  oo 

9.4281 

9.7614 

IO.OOOO 

10.2386 

10.5719 

eo 

1 

7.6398 

301 

4356 

7 

7658 

4 

0038 

4 

2429 

4 

5796 

8 

59 

2 

7.9409 

176 

443° 

14 

7701 

9 

0076 

8 

2474  i      9 

5873 

16 

58  i 

3 

8.1170 

125 

45°3 

21 

7745       13 

0114 

ii 

2518  f    13 

5952 

25 

57 

4 

8.2419 

97 

4575 

28 

7788 

17 

0152 

15 

2562       18 

6032 

33 

56 

5 

8.3389  !     79 

9.4646 

35 

9-7831 

21 

10.0190       19 

10.2607         22 

10.6114 

41 

55 

6 

4181 

67 

y- 

47l6 

42 

7873 

26 

0228       23 

2652          27 

6196 

49 

54 

7 

4851 

58 

4785 

49 

7916 

30 

0265  1   27 

2698  ' 

31 

6281 

57 

53 

8 

5431 

51 

4853             56 

7958 

34 

°3°3  i    3° 

2743 

36 

6366 

66 

52 

9 

5943 

46 

4921             63 

8000       39 

0341 

34 

2789 

40 

6454 

74 

51 

10 

8.6401 

41 

94987 

9.8042 

10.0379 

10.2835 

10.6542 

50 

11 

6815 

38 

5053 

6 

8084         4 

0418 

4 

2882 

5 

6633 

10 

49 

12 

7194 

35 

5Il8 

13 

8125        .8 

0456 

8 

2928 

9 

6725 

20 

48 

i  13 

7542 

32 

5l82 

19 

8167 

12 

0494 

ii 

2975 

13 

6819 

29 

47 

14 

7865 

30 

5245 

25 

8208 

16 

15 

3023 

19 

6915 

39 

46  ! 

;  1$ 

8.8165 

28 

9.5308 

31 

9.8249 

20 

10.0570 

19 

10.3070 

24 

10.7013 

49 

45 

16 

8446 

26 

5370 

38 

8290 

25 

0608 

23 

3Il8 

28 

7H3 

59 

44 

17 

8711 

25 

543  * 

44 

8331 

29 

0647 

27 

3l66 

33 

7216 

69 

43 

18 

8960 

24 

549  ! 

50 

8371 

33 

0685 

30 

3215 

38 

7320 

78 

42 

19 

9196 

22 

5551 

56 

8412 

37 

0723 

34 

3264 

43 

7427 

88 

41 

20 

8.9420 

21 

9.5611 

9.8452 

10.0762 

10.3313 

10-7537 

40 

21 

9633 

20 

5669 

•6 

8493 

4 

0800 

4 

3363 

5 

7649 

13 

39 

22 

9836 

19 

5/27 

ii 

8533         8 

0839 

8 

3413 

10 

7764 

26 

38 

23 

9.0030 

19 

5785 

17 

8573        I2 

0878 

12 

3463 

15 

7882 

39 

37 

24 

0216 

18 

5842 

22 

8613 

16 

0916 

16 

35H 

20 

8003 

52 

36 

25 

9-0395 

•  17 

9.5898 

28 

9-8652 

20 

10.0955 

19 

10.3565 

25 

10.8127 

65 

35 

>  26 

0567 

16 

5954 

34 

8692 

24 

0994 

23 

3617 

31 

8255 

78 

34 

27 

0732 

16 

6009 

39 

8732 

28 

1033 

27 

3669 

36 

8387 

91 

33 

28 

0891 

15 

6064 

45 

8771 

32 

1072 

31 

3721 

8522 

104 

32 

29 

1045 

6118 

50 

88n 

36 

mi 

35 

3774 

46 

8662 

1  1  7 

31 

30 

9.1194 

9-6172 

9.8850 

10.1150 

10.3828 

10.8806 

14 

30 

31 

1338 

J3 

6226 

5 

8889 

4 

1189 

4 

3882 

6 

8955 

15 

29 

32 

1478 

26 

6279 

10 

8928 

8 

1229 

8 

3936 

n 

9109 

15 

28 

33 

1613 

39 

6331 

16 

8967 

12 

1268 

12 

17 

9268 

16 

27 

34 

'745 

52 

6383 

21 

9006 

16 

1308 

16 

4046 

22 

9433 

16 

26 

35 

9.1873 

65 

9-6435 

26 

9.9045 

19 

10.1348 

20 

10.4102 

28 

10.9605 

17 

25 

36 

1997 

78 

6486 

31 

9084. 

23 

1387 

24 

4158 

34 

9784 

18 

24 

37 

2118 

6537 

36 

9122 

27 

1427 

28 

4215 

39 

9970 

19 

23 

38 

2236 

104 

6587 

42 

9161 

31 

1467 

32 

4273 

45 

11.0164 

19 

22  1 

39 

2351 

117 

6637 

47 

9200 

35 

1507 

36 

433  i 

50 

0367 

20 

21 

40 

9.2463 

9.6687 

9-9238 

10.1548 

10.4389 

11.0580 

21 

20 

41 

2573 

10 

6736 

5 

9277 

4 

1588 

4 

4449 

6 

0804 

22 

19 

42 

2680 

20 

6785 

10 

9315 

8 

1629 

8 

45°9 

12 

1040 

24 

18  i 

43 

2784 

29 

6834 

14 

9353 

n 

1669 

12 

4569 

19 

1289 

25 

17 

44 

2887 

39 

6882 

19 

9392 

15 

1710 

16 

4630 

25 

1554 

26 

16 

45 

9-2987 

49 

9.6930 

24 

19 

10.1751 

20 

10.4692 

31 

11-1835 

28 

15 

46 

3085 

59 

6977 

29 

9468 

23 

1792 

25 

4755 

37 

2135 

3° 

14 

^47 

3181 

7025 

34 

9506 

27 

1833 

29 

4818 

43 

2458 

32 

13 

48 

3275 

78 

7072 

38 

9544 

3° 

1875 

33 

4882 

50 

2806 

35 

12 

49 

3367 

88 

7118 

43 

9582 

34 

1916 

37 

4947 

56 

3185 

38 

11 

50 

9-3458 

9-7165 

9.9621 

10.1958 

10.5013 

n-3599 

41 

10  i 

51 

3546 

8 

7211 

5 

9659 

4 

2000 

4 

5079 

7 

4057 

46 

9 

52 

3634 

16 

7257 

9 

9697 

8 

2O42 

8 

5H7 

14 

4569 

5J 

8 

53 

3719 

25 

73°2 

13 

9735 

n 

2084 

13 

5215 

21 

5I5° 

58 

7 

54 

3804 

33 

7348 

18 

9773 

15 

2127 

17 

5284 

28 

5819 

67 

6 

55 

9-3886 

9-7393 

22 

9.9810 

19 

I0.2I69 

21 

10-5354 

35 

11.6611 

79 

5 

56 

3968 

49 

7438 

27 

9848 

23 

2212 

25 

5425 

43 

7581 

97 

4 

57 

4048 

57 

7482 

31 

9886 

27 

2255 

3° 

5497 

5° 

8830 

'25 

3 

58 

4127 

66 

7526 

36 

9924 

30 

2299 

5570 

57 

12.0591 

176 

2 

59 

4204 

74 

757i 

40 

9962 

34 

2342 

38 

5644 

64 

3602 

301 

1 

60 

9.4281 

9.7614 

10.0000 

10.2386 

10.5719 

+  00 

0 

llh 

Diff 

ioh 

Diff 

9h 

Diff 

8h 

Diff 

7h 

Diff 

6h 

Diff. 

m 

5h 

O.ml 

4h 

Om.l 

3" 

Om.l 

2h 

O'u.l 

lh 

0».l 

Oh 

Om.l 

1X1 

jPpRRead  "Tangents  "  in  upper  line  and  "Cotangents"  in  lower  line  of  Hour  Arguments  alike  at  top  and  bottom, 
taking  the  minutes  in  left  or  right  column  according  as  the  hours  are  taken  at  top  or  bottom. 

TABLE  XIV. 


79 


Degrees.  ' 

1 

00 
0 

LengtBis  of  Circular  Arcs. 

6 

O°.2 
12' 

0°3 

18' 

0°.4 
24' 

0°.5 
30' 

0°.6 
36' 

0°.7 
42' 

0°.8 

48' 

0°.9 
54' 

O 

o.oooo   0.0017 

0.0035 

0.0052 

0.0070 

0.0087 

0.0105 

0.0122 

0.0140 

0.0157 

1 

0175    0192 

0209 

0227 

0244 

0262 

0280 

0297 

0317 

0332 

2 

0349    0367 

0384 

0401 

0419 

0436 

0454 

0471 

0489 

0506 

3 

0524    0541 

°559 

0576 

0594 

0611 

0628 

0646 

0664 

0681 

4 

0698    0716 

0733 

0750 

0768 

0785 

0803 

0820 

0838 

0855 

5 

0.0873   0.0890 

0.0908 

0.0925 

0.0943 

0.0960 

0.0978 

0.0995 

O.IOI2 

0.1030 

6 

1047    1065 

1082 

1099 

1117 

H34 

1152 

1169 

Il87 

1204 

7 

1222      1239 

1257    1274 

1292 

1309 

1327 

1344 

I36l 

1379 

8 

1396      1414 

1431    1449 

1466 

1484 

1501 

I5l8 

1536 

9 

1571      1588      1606      1623 

1641 

1658 

1676 

1693 

I7IO 

1728 

10 

0.1745    0.1763!   0.1780    0.1798 

0.1815 

0.1833 

0.1850 

0.1867 

0.1885 

0.1902 

11 

1920      1937      1955  |     1972 

1990 

2007 

2025 

2042 

2059 

2077 

12 

2094      2112      2129  !    2147 

2164 

2182    2199 

22l6     2234 

2251 

13 

2269      2286      2304  i    2321 

2339 

2356    2374  f   2391    2409    2426 

14 

2443 

2461    2478  j   2496 

2513 

253!    2548    2566 

2583     26OI 

15 

0.26l8 

0.2635   0.2653  j  0.2670 

0.2688 

0.2705   0.2723   0.2740 

0.2758   0.2775 

16 

2793 

2810    2827  |   2845 

2862 

2880  i   2897    2915  I   2932 

2950 

17 

2967 

2985  3002  1  3019  3037 

3054   3072    3089  !   3107 

3124 

18 

3H2 

3159  i  3177  3194  3212 

3229    3246  |   3264   3281 

3299 

19 

3316 

3334  i   335  '    3368    33g6 

3403    3421  |   3438 

3456 

3473 

20 

0-3491 

0.3508 

0.3526 

0-3543 

0.3561 

0.3578 

0.3596 

0.3613 

0.3630 

0.3648 

21 

3665 

3683 

37oo 

37i8 

3735 

3752 

3770 

3787 

3805 

3822 

22 

3840 

3857 

3875 

3892 

3910 

3927 

3945 

3962 

3979 

3997 

23 

4014 

4032 

4049 

4067 

4084 

4102 

4119 

4136 

4154 

4171 

24 

4189 

4206 

4224 

4241 

4259 

4276 

4294 

4311 

4328 

4346 

25 

0.4363 

0.4381 

0.4398 

0.4416 

0.4433 

0.4451 

0.4468 

0.4486 

0-4503 

0.4520 

26 

4538  j   4555 

4573 

4590 

4608 

4625 

4643 

4660 

4678 

4695 

27 

4712  i   473° 

4747 

4765    4782 

4800 

4817 

4835 

4852 

4869 

28 

4887  !   4904 

4922 

4939 

4957 

4974 

4992 

5009 

5027 

5044 

29 

5061    5079 

5096 

5"4 

5*49 

5166 

5184 

5201 

5219 

30 

0.5236   0.5253 

0.5271 

0.5288 

0.5306 

0.5323 

0.5341 

0.5358 

0.5376 

0-5393 

31 

5411    5428 

5445 

5463 

548o 

5498 

5515 

5533 

5550 

5568 

32 

5585    5603 

5620 

5638 

5655 

5673 

5690 

5707 

5725 

5743 

33 

576o    5777 

5795 

5812 

5830 

5847 

5865 

5882 

5899 

5917 

34 

5934 

5952 

5969 

5986 

6004 

6021 

6039 

6056 

6074 

6091 

35 

0.6109 

0.6126 

0.6144 

0.6161 

0.6179 

0.6196 

0.6213 

0.6231 

0.6248 

0.6266 

36 

6283 

6301 

6318 

6336 

6353 

6370 

6388 

6405 

6423 

6440 

37 

6458 

6475 

6493 

6510 

6528 

6545 

6563 

6580 

6597 

6615 

38 

6632 

6650 

6667 

6685 

6702 

6720 

6737 

6754 

6772 

6789 

39 

6807 

6824 

6842 

6859 

6877 

6894 

6912 

6920 

6946 

6964 

40 

0.6981 

0.6999 

0.7016 

0.7034 

0.7051 

0.7069 

0.7086 

0.7103 

0.7121 

0.7138 

41 

7156 

7173 

7191 

7208 

7226 

7243 

7261 

7278 

7296 

7313 

42 

733° 

7348 

7365 

7383 

7400 

7418 

7435 

7453 

7470 

7487 

43 

75°5 

7522 

7540 

7557 

7575 

7592 

7610 

7627 

7645 

7662 

44 

7679 

7697 

77H 

7732 

7749 

7767 

7784 

7802 

7819 

7836 

45 

0.7854 

0.7871 

0.7889 

0.7906 

0.7924 

0.7941 

0-7959 

0.7976 

0.7994 

0.8011 

46 

8029 

8046 

8063 

8081 

8098 

8116 

8133 

8151 

8168 

8186 

47 

8203 

8220 

8238 

8255 

8273 

8290 

8308 

8325 

8343 

8360 

48 

8378 

8395 

8413 

8430 

8448 

8465 

8482 

8500 

8517 

8535 

49 

8552 

8570 

8587 

8604 

8622 

8639 

8657 

8674 

8692 

8709 

50 

0.8727 

0.8744 

0.8762 

0-8779 

0.8797 

0.8814 

0.8831 

0.8849 

0.8866 

0.8884 

51 

8901 

8919    8936 

8953 

8971 

8988 

9006 

9023 

9041 

9058 

52 

9076 

9093  ;   9111 

9128 

9146 

9163 

9181 

9198 

9215 

9233 

53 

9250 

9268 

9285 

93°3 

9320 

9338 

9355 

9372 

9390 

9407 

54 

9425 

*  9442 

9460 

9477 

9495 

9512 

9530 

9547 

9564 

9582 

55 

0.9599 

0.9617 

0.9634 

0.9652 

0.9669 

0.9687 

0.9704 

0.9721 

0.9739 

0.9756 

56 

9774 

9791 

9809 

9826 

9844 

9861 

9879 

9896 

9914 

993  i 

57 

9948 

9966 

9983 

1.  000  1 

1.0018 

1.0036 

1-0053 

1.0071 

i.  0088 

1.0105 

58 

1.0123 

1.0140 

1.0158 

0175 

0193 

02  10 

0228 

0245 

0263 

0280 

59 

0297 

0315 

0332 

0350 

0367 

0385 

0402 

0420 

0437 

0455 

1 

I 

80 


TABLE  XV. 


Natural  Suit's. 

Degrees. 

o°.o 

0 

6' 

0°.2 
12' 

0°.3 

18'' 

0°.4 
24' 

0°.5 
30' 

0°.6 
36' 

0°.7 
42' 

0=.8 

48' 

0^.9   1  .0 
54'   60' 

Degrees. 

0 

o.oooo 

0.0017 

0.0035 

0.0052 

0.0070 

0.0087 

0.0105 

0.0122 

0.0140 

0.0157 

0.0175 

89 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

03H 

0332 

0349 

88 

2 

0349 

0366 

0384 

0401 

0419 

0436 

0454 

0471 

0488 

0506 

0523 

87 

3 

0523 

0541 

0558 

0576 

0593 

0610 

0628 

0645 

0663 

0680 

0698 

86 

4 

0698 

0715 

0732 

075° 

0767 

0785 

0802 

0819 

0837 

0854 

0872 

85 

5 

0.0872 

0.0889 

0.0906 

0.0924 

0.0941 

0.0958 

0.0976 

0.0993 

O.IOII 

0.1028 

0.1045 

84 

6 

1045 

1063 

1080 

1097 

1115 

1132 

1149 

1167 

1184 

1201 

1219 

83 

7 

1219 

1236 

1253 

1271 

1288 

1305 

1323 

1340 

1357   1374 

1392 

82 

8 

1392 

1409 

1426 

1444 

1461 

1478 

1495 

1513 

1530 

1547 

1564 

81 

9 

1582 

1599 

1616 

1633 

1650 

1668 

1685 

1702 

1719 

i/36 

80 

10 

0.1736 

o.i754 

0.1771 

0.1788 

0.1805 

0.1822 

0.1840 

0.1857 

0.1874 

0.1891 

0.1908 

79 

11 

1908 

1925 

1942 

1959 

1977 

1994 

2OII 

2028 

2045 

2O62 

2079 

78 

12 

2079 

2096 

2113 

2130 

2147 

2164 

2181 

2198 

2215 

2233 

2250 

77  i 

13 

2250 

2267 

2284 

2300 

2317 

2334 

2351 

2368 

2385 

2402 

2419 

76 

14 

2419 

2436 

2453 

2470 

2487 

2504 

2521 

2538 

2554 

2571 

2588 

75 

15 
16 

0.2588 
2756 

0.2605 

2773 

0.2622 
2790 

0.2639 
2807 

0.2656 
2823 

0.2672 
2840 

0.2689 
2857 

0.2706 

2874 

0.2723 
2890 

0.2740 
2907 

0.2756 
2924 

74 
73 

17 

2924 

2940 

2957 

2974 

2990 

3007 

3024 

3040 

3057 

3°74 

3090 

72 

18 

3090 

3107 

3123 

3156 

3173 

3190 

3206 

3223 

3239 

3256 

71  ! 

19 

3256 

3272 

3289 

33°5 

3322 

3338 

3355 

3371. 

3387 

3404 

3420 

7O 

2O 

0.3420 

0-3437 

0-3453 

0.3469 

0.3486 

0.3502 

0.3518 

0-3535 

0.3551 

o-3567 

0.3584 

69 

21 

22 

3584 
3746 

3600 
3762 

3616 
3778 

3633 
3795 

3649 
3811 

3665 
3827 

3681 
3843 

3697 
3859 

37H 
3875 

3730 
3891 

3746 
3907 

68 
67 

23 

3907 

3923 

3939 

3955 

397i 

3987 

4003 

4019 

4035 

4051 

4067 

66 

24 

4067 

4083 

4099 

4H5 

4I31 

4H7 

4163 

4179 

4195 

4210 

4226 

65 

25 

0.4226 

0.4242 

0.4258 

0.4274 

0.4289 

0.4305 

0.4321 

0-4337 

0.4352 

0.4368 

0.4384 

64 

26 

4384 

4399 

4415 

443  * 

4446 

4462 

4478 

4493 

45°9 

4524 

4540 

63 

27 

4540 

4555 

4571 

4586 

4602 

4617 

4633 

4648 

4664 

4679 

4695 

62 

28 
29 

4848 

4710 
4863 

4726 
4879 

4741 
4894 

4756 
4909 

4772 
4924 

4787 
4939 

4802 
4955 

4818 
4970 

4833 
4985 

5000 

61 

60 

30 

0.5000 

0.5015 

0.5030 

0.5045 

0.5060 

0.5075 

0.5090 

0.5105 

0.5120 

0.5135 

0.5150 

59 

31 

5150 

5165 

5180 

5*95 

5210 

5225 

5240 

5255 

5270 

5284 

5299 

58 

32 

5299 

53*4 

5329 

5344 

5358 

5373 

5388 

5402 

5417 

5432 

5446 

57 

33 

34 

5446 
5592 

5606 

5476 
5621 

5490 
5635 

5505 

5519 
5664 

5534 
5678 

5548 
5693 

5563 
5707 

5577 
572i 

5592 
5736 

56 

55  I 

35 
36 

0.5736 
5878 

o.575o 
5892 

0.5764 
5906 

0-5779 
5920 

0-5793 
5934 

0.5807 

0.5821 
5962 

0-5835 
5976 

0.5850 
5990 

0.5864 
6004 

0.5878 
6018 

54 
53 

37 

6018 

6032 

6046 

6060 

6074 

6088 

6101 

6115 

6129 

6143 

6157 

52 

38 

6157 

6170 

6184 

6198 

6211 

6225 

6239 

6252 

6266 

6280 

6293 

51 

39 

6293 

6307 

6320 

6334 

6347 

6361 

6374 

6388 

6401 

6414 

6428 

50 

40 

0.6428 

0.6441 

0.6455 

0.6468 

0.6481 

0.6494 

0.6508 

0.6521 

0.6534 

0.6547 

0.6561 

49 

41 

6561 

6574 

6587 

6600 

6613 

6626 

6639 

6652 

6665 

6678 

6691 

48  ; 

42 

6691 

6704 

6717 

6730 

6743 

6756 

6769 

6782 

6794 

6807 

6820 

47  ! 

43 

6820 

6833 

6845 

6858 

6871 

6884 

6896 

6909 

6921 

6934 

6947 

46 

44 

6947 

6959 

6972 

6984 

6997 

7009 

7022 

7034 

7046 

7059 

7071 

45  : 

i 

60' 

54' 

48' 

42' 

36' 

30' 

24' 

18' 

12' 

6 

0' 

1 

1 

ro 

0°.9 

0°.8 

O°.7 

0D.6 

0°.5 

04 

03 

0°.2 

01 

o°.o 

60 

5  ! 

Natural  Cosines. 

TABLE  XV. 


81 


1 

Natural  Sim**. 

co 
0) 

0.0 
O 

0°.l 

6 

Oa.2 
12 

0°.3 

18' 

0°.4 
24' 

0°.5 
3O' 

0°.6 
36' 

0°.7 
42 

0°.8 

48' 

0°.9 
54 

1°.0 

60 

egrees. 

1 

Q  i 

45 

0.7071 

0.7083 

0.7096 

0.7108 

0.7120 

0.7133 

0.7145 

0.7157 

0.7169 

0.7181 

0.7193 

44 

46 

7i93 

7206 

7218 

7230 

7242 

7254 

7266 

7278 

7290 

7302 

73H 

43 

47 

73  H 

7325 

7337 

7349 

736i 

7373 

7385 

7396 

7408 

7420 

743  i 

42 

48 

743  r 

7443 

7455 

7466 

74/8 

7490 

7501 

7513 

7524 

7536 

7547 

41 

49 

7547 

7559 

7570 

758i 

7593 

7604 

'  7615 

7627 

7638 

7649 

7660 

40 

50 

0.7650 

0.7672 

0.7683 

0.7694 

0.7705 

0.7716 

0.7727 

0.7738 

0.7749 

0.7760 

0.7771 

39 

51 

777i 

7782 

7793 

7804 

781  S 

7826 

7837 

7848 

7859 

7869 

7880 

38 

*i 

7880 

7891 

7902 

7912 

7923 

7934 

7944 

7955 

7965 

7976 

7986 

37 

53 

7986 

7997 

8007 

8018 

8028 

8039 

8049 

8059 

8070 

8080 

8090 

36 

54 

8090 

8100 

8m 

8121 

8131 

8141 

8151 

8161 

8171 

8181 

8192 

35 

55 

0.8192 

0.8202 

0.8211 

0.8221 

0.8231 

0.8241 

0.8251 

0.8261 

0.8271 

0.8281 

0.8290 

34 

50 
57 

8293 

83^7 

8300 
8396 

8310 
8406 

8320 
8415 

8329 
8425 

8339 
8434 

8348 
8443 

8358 
8453 

8368 
8462 

8377 
8471 

8387 
8480 

33 
32 

5§ 
59 

8480 
8572 

8490 
8581 

8499 
8590 

8508 
8599 

8517 
8607 

8526 
8616 

8536 
8625 

8545 
8634 

8554 
8643 

8563 
8652 

8572 
8660 

31  1 
30 

69 

o.865o 

0.8669 

0.8678 

0.8686 

0.8695 

0.8704 

0.8712 

0.8721 

0.8729 

0.8738 

0.8746 

29 

61 

8746 

8755 

8763 

8771 

8780 

8788 

8796 

8805 

8813 

8821 

8829 

28 

0i 

8829 

8838 

8846 

8854 

8862 

8870 

8878 

8886 

8894 

8902 

8910 

27 

63 
64 

8910 
8988 

8918 
8996 

8926 
9003 

8934 
9011 

8942 
9018 

8949 
9026 

8957 
9033 

8965 
9041 

8973 
9048 

8980 
9056 

8988 
9063 

26 
25 

65 

0.9063 

0.9070 

0.9078 

0.908-5 

0.9092 

0.9100 

0.9107 

0.9114 

0.9121 

0.9128 

o.9i35 

24 

66 

9135 

9143 

9i50 

9157 

9164 

9171 

9178 

9184 

9191 

9198 

9205 

23  i 

67 

9205 

9212 

9219 

9225 

9232 

9239 

9245 

9252 

9259 

9265 

9272 

22  ! 

68 

9272 

9278 

9285 

9291 

9298 

93°4 

9311 

93J7 

9323 

933° 

9336 

21  ! 

69 

9336 

9342 

9348 

9354 

9361 

9367 

9373 

9379 

9385 

939i 

9397 

20 

70 

0-9397 

0.9403 

0.9409 

0.9415  ' 

0.9421 

0.9426 

0.9432 

0.9438 

0.9444 

0.9449 

0-9455 

19 

71 

9455 

9461 

9466 

9472 

9478 

9483 

9489 

9494 

9500 

9505 

95" 

18 

72 
73 
74 

95" 
9563 
9613 

95i6 
9568 
9617 

952i 

9573 
9622 

9527 
9578 
9627 

9532 

9583 
9632 

9537 
9588 
9636 

9542 
9593 
9641 

954f 
9598 
9646 

9553 
9603 
9650 

9558 
9608 

9655 

9563 
9613 

9659 

17 
16 
15 

75 

0.9659 

0.9664 

0.9668 

0.9673 

0.9677 

0.9681 

0.9686 

0.9690 

0.9694 

0.9699 

0.9703 

14 

76 

97°3 

9707 

9711 

97i5 

9720 

9724 

9728 

9732 

9736 

9740 

9744 

13 

77 

9744 

9748 

975i 

9755 

9759 

9763 

9767 

9770 

9774 

9778 

9781 

12 

78 

9781 

9785 

9789 

9792 

9796 

9799 

9803 

9806 

9810 

9813 

9816 

11 

79 

9816 

9820 

9823 

9826 

9829 

9833 

9836 

9839 

9842 

9845 

9848 

1O 

SO 

0.9848 

0.985  1 

0.9854 

0.9857 

0.9860 

0.9863 

0.9866 

0.9869 

0.9871 

0.9874 

0.9877 

9 

81 

9877 

9880 

9882 

9885 

9888 

9890 

9893 

9895 

9898 

9900 

9903 

8 

82 

9903 

99°5 

9907 

9910 

9912 

9914 

9917 

9919 

992i 

9923 

9925 

7 

83 

9925 

9928 

993° 

9932 

9934 

9936 

9938 

9940 

9942 

9943 

9945 

6  i 

84 

9945 

9947 

9949 

995  i 

9952 

9954 

9956 

9957 

9959 

9960 

9962 

5 

85 

0.9962 

0.9963 

0.9965 

0.9966 

0.9968 

0.9969 

0.9971 

0.9972 

0.9973 

0.9974 

0.9976 

4 

86 

9976 

9977 

9978 

9979 

9980 

998i 

9982 

9983 

9984 

9985 

9986 

3 

87 

9986 

9987 

9988 

9989 

9990 

9990 

9991 

9992 

9993 

9993 

9994 

2 

88 

9994 

9995 

9995 

9996 

9996 

9997 

9997 

9997 

9998 

9998 

9998 

1 

89 

9998 

9999 

9999 

9999 

9999 

I.OOOO 

I.OOOO 

I.OOOO 

I.OOOO 

i  .0000 

I.OOOO 

O 

1 

60' 

54 

48' 

12- 

36' 

3O' 

24' 

18' 

12 

6 

0' 

i 

a 
i 

1°.O 

O°.9 

0°.8 

0°.7 

0°.6 

0°.5 

0°.4 

0°.3 

0°2 

0^.1 

O°.O 

1 

Natural  Cosines. 

82 


TABLE  XVI. 


Natural  Tangents. 

Degrees. 

o°.o 

0 

6' 

O°.2 
12' 

O°.3 

18' 

0°.4 
24' 

0°.5 
30' 

0  .6 
36 

0.7 
42' 

0°.8 

48' 

0°.9 
54' 

1  .0 
6O 

Degrees.  | 

0 

0.0000 

0.0017 

0.0035 

0.0052 

0.0070 

0.0087 

0.0105 

O.OI22 

0.0140 

0.0157 

0.0175 

89 

1 

0175 

0192 

0209 

0227 

0244 

0262 

0279 

0297 

0314 

0332 

°349 

88 

2 

0349 

0367 

0384 

0402 

0419 

0437. 

0454 

0472 

0489 

0507 

0524 

87 

3 

0524 

0542 

0559 

0577 

0594 

0612 

0629 

0647 

0664 

0682 

0699 

86  ; 

4 

0699 

0717 

0734 

0752 

0769 

0787 

0805 

0822 

0840 

0857 

0875 

85 

5 

0.0875 

0.0892 

0.0910 

0.0928 

0.0945 

0.0963 

0.0981 

0.0998 

0.1016 

0.1033 

0.1051 

84 

6 

1051 

1069 

1086 

1104 

1  122 

"39 

"57 

"75 

1192 

1210 

1228 

83 

7 

1228 

1246 

1263 

1281 

1299 

1317 

1334 

1352 

1370 

1388 

1405 

82 

8 

1405 

1423 

1441 

H59 

M77 

1495 

1512 

1530 

1548 

1566 

1584 

81 

9 

1584 

1602 

1620 

1638 

1655 

1673 

1691 

1709 

1727 

1745 

1763 

80 

10 

0.1763 

0.1781 

0.1799 

0.1817 

0.1835 

0.1853 

0.1871 

0.1890 

0.1908 

0.1926 

0.1944 

79 

11 

1944 

1962 

1980 

1998 

2Ol6 

2035 

2053 

2071 

2089 

2107 

2126 

78 

12 

2126 

2144 

2162 

2180 

2199 

2217 

2235 

2254 

2272 

2290 

2309 

77  ! 

13 

2309 

2327 

2345 

2364 

2382 

2401 

2419 

2438 

2456 

2475 

2493 

76 

14 

2493 

2512 

2530 

2549 

2568 

2586 

2605 

2623 

2642 

266l 

2679 

75 

15 
16 

0.2679 
2867 

0.2698 
2886 

0.2717 
2905 

0.2736 
2924 

0.2754 
2943 

0.2773 
2962 

0.2792 
2981 

0.2811 
3000 

0,2830 
3019 

0.2849 
3038 

0.2867 
3°57 

74 
73 

17 

3057 

3076 

3096 

3"5 

3134 

3153 

3172 

3*91 

3211 

3230 

3249 

72 

18 

3249 

3269 

3288 

33°7 

3327 

3346 

3365 

3385 

3404 

3424 

3443 

71 

19 

3443 

3463 

3482 

3502 

3522 

3541 

356i 

358i 

3600 

3620 

3640 

7O 

2O 

0.3640 

0-3659 

0.3679 

0.3699 

0.3719 

0-3739 

0-3759 

0-3779 

0-3799 

0.3819 

0-3839 

69 

2  1 

3839 

3859 

3879 

3899 

3919 

3939 

3959 

3979 

4000 

4O2O 

4040 

68 

22 

4040 

4061 

4081 

4101 

4122 

4142 

4163 

4183 

4204 

4224 

4245 

67 

23 

4245 

4265 

4286 

43°7 

4327 

4348 

4369 

4390 

44" 

443  * 

4452 

66 

24 

4452 

4473 

4494 

4515 

4536 

4557 

4578 

4599 

4621 

4642 

4663 

65 

i  25 

0.4663 

0.4684 

0.4706 

0.4727 

0.4748 

0.4770 

0.4791 

0.4813 

0.4834 

0.4856 

0.4877 

64 

26 

4877 

4899 

4921 

4942 

4964 

4986 

5008 

5029 

5051 

5°73 

5°95 

63 

27 

5°95 

5117 

5J39 

5161 

5^4 

5206 

5228 

5250 

5272 

5295 

5317 

62 

28 

53  *  7 

5340  !  5362 

5384 

540.7 

5430 

5452 

5475 

5498 

5520 

5543 

61 

29 

5543 

5566 

5589 

5612 

5635 

5658 

5681 

5704 

5727 

5750 

5774 

60 

30 
31 

0-5774 
6009 

0-5797 
6032 

0.5820 
6056 

°ios2 

0.5867 
6104 

0.5890 
6128 

0.5914 
6152 

0.5938 
6176 

0.5961 
6200 

0.5985 
6224 

0.6009 
6249 

59 

58 

32 
33 
34 

6249 
6494 
6745 

6273 
6519 
6771 

6297 

6544 
6796 

6322 
6569 
6822 

6346 
6594 
6847 

6371 
6619 
6873 

6395 
6644 
6899 

6420 
6669 
6924 

6445 
6694 

6950 

6469 
6720 
6976 

6494 
6745 
7002 

57 
56 
55 

35 

0.7002 

0.7028 

0.7054 

0.7080 

0.7107 

0-7133 

0.7159 

0.7186 

0.7212 

0.7239 

0.7265 

54 

i  36 

7265 

7292 

7319 

7346 

7373 

7400 

7427 

7454 

748i 

7508 

7536 

53 

37 

7536 
7813 

7563 
7841 

7590 
7869 

7618 
7898 

7646 
7926 

7673 
7954 

7701 
7983 

7729 
8012 

7757 
8040 

7785 
8069 

7813 
8098 

52 
51 

j  39 

8098 

8127 

8156 

8185 

8214 

8243 

8273 

8302 

8332 

8361 

8391 

5O 

40 

0.8391 

0.8421 

0.8451 

0.8481 

0.8511 

0.8541 

0.8571 

0.8601 

0.8632 

0.8662 

0.8693 

49 

41 

8693 

8724 

8754 

8785 

8816 

8847 

8878 

8910 

8941 

8972 

9004 

48 

|42 

9004 

9036 

9067 

9099 

9131 

9163 

9195 

9228 

9260 

9293 

9325 

47 

43 

9325 

9358 

9391 

9424 

9457 

9490 

9556 

9590 

9623 

9657 

46 

i  44 

9657 

9691 

9725 

9759 

9793 

9827 

9861 

9896 

9930 

9965 

I.OOOO 

45 

i 

6O' 

54' 

48' 

42' 

36' 

30' 

24' 

18' 

12' 

6' 

0 

i 

s 

!•-• 

O°.9 

0.8 

0°.7 

0°.6 

0°.5 

0°.4 

0°.3 

O°.2 

01 

Oc.O 

I 
3 

Natural  Cotangents. 

TABLE  XVI. 


83 


Natural  Tangents. 

Degrees. 

o°.o 

0 

6' 

O°.2 
12' 

0°.3 

18' 

0°.4 
24' 

0°.5 
3O' 

0°.6 

36 

0°.7 
42' 

0°.8 

48' 

0°.9 
54' 

60' 

Degrees. 

45 

1.  0000 

1-0035 

1.0070 

1.0105 

1.0141 

1.0176 

I.O2I2 

1.0247 

1.0283 

1.0319 

1-0355 

44 

46 

0355 

0392 

0428 

0464 

0501 

0538 

°575 

0612 

0649 

0686*     0724 

43 

j  47 

0724 

0761 

0799 

0837 

0875 

0913 

095  1 

0990 

1028 

1067 

1106 

42 

IN 

1106 

H4S 

1184 

1224 

1263 

1303 

1343 

1383 

1423 

1463- 

1504 

41 

49 

1504 

1544 

1585 

1626 

1667 

1708 

1750 

1792 

1833 

1875 

1918 

40 

50 

1.1918 

1.1960 

I.2OO2 

1.2045 

1.2088 

1.2131 

I.2I74 

1.2218 

1.2261 

1-2305 

1.2349 

39 

51 

2349 

2393 

2437 

2482 

2527 

2572 

2617 

2662 

2708 

2753 

2799 

3S 

52 

2799 

2846 

2892 

2938 

2985 

3032 

3079 

3127 

3175 

3222 

3270 

37 

53 

3270 

33*9 

3367 

3465 

3514 

3564 

3613 

3663 

3713 

3764 

36 

54 

3764 

3865. 

3916 

3968 

4019 

4071 

4124 

4176 

4229 

4281 

35 

55 

1.4281 

1-4335 

1.4388 

1.4442 

1.4496 

1-4550 

1.4605 

1.4660 

.  I.47I5 

1.4770 

1.4826 

34 

56 

4826 

4882 

4938 

499.4 

5051 

5108 

5l66i          5224 

5282 

5340 

5399 

33 

57 

5399 

545S 

55'7 

5577 

5637 

5697 

5757 

5818 

5880 

5941 

6003 

32 

58 

6003 

6066 

6128 

6191 

6255 

6319 

6383 

6447)       6512 

6577 

6643 

31 

59 

6643      6709 

6775 

6842 

6909 

6977 

7045 

7182 

7251 

7321 

30 

60 

I-732i|  I-7391 

1.7461 

I-7532 

1.7603 

I-7675 

1-7747 

1.7820 

1-7893 

1.7966 

1.8040 

29 

61 

8040'     81115 

8190 

8265 

8341 

8418 

8495 

8572!       8650 

8728 

8807 

28 

62 

8807 

8887 

8967 

9047 

9128 

9210 

9292 

9375 

9458 

95421      9626 

27 

63 

9626 

9711 

9797 

9883 

9970 

2.0057 

2.0145 

2.0233 

2.0323 

2.0413)  2.0503 

26 

64 

2.0503 

2.0594 

2.0686 

2.0778 

2.0872 

0965 

1060 

"55 

1251 

1348 

1445 

25 

65 

2.1445 

2.1543 

2.1642 

2.1742 

2.1842 

2.1943 

2.2045 

2.2148 

2.2251 

2.2355 

2.2460 

24 

66 

2460 

2566 

2673 

2781 

2889 

2998 

3109 

3220 

3332 

3445 

3559 

23 

67 

3559 

3673 

3789 

39o6 

4023 

4142 

4262 

4383 

4504 

4627 

22 

68 

475i 

4876 

5002 

5I29 

5257 

5386 

5517 

5649 

5782 

59i6 

6051 

21 

69 

6051 

6187 

6325 

6464 

6605 

6746 

6889 

7034 

7179 

7326 

7475 

20 

7O 

2-7475 

2.7625 

2.7776 

2.7929 

2.8083 

2.8239 

2.8397 

2.8556 

2.8716 

2.8878 

2.9042 

19 

71 

9042 

9208 

9375 

9544 

97H 

9887 

3.0061 

3.0237 

3-0415 

3.0595 

3.0777 

18 

72 
73 

3.0777 
2709 

3.0961 
2914 

3.1146 
3122 

3-1334 
3332 

3-1524 
3544 

3.1716 
3759 

1910 
3977 

2106 

2305 
4420 

2506 
4646 

2709 
4874 

17 
16 

74 

4874 

5105 

5339 

5576 

5816 

6059 

6305 

6554 

6806 

7062 

15 

75 

3-7321 

3-7583 

3.7848 

3.8118 

3-839I 

3.8667 

3.8947 

3.9232 

3-9520 

3-9812 

4.0108 

14 

76 

4.0108 

4.0408 

4-0713 

4.1022 

4.1335 

4-1653 

4.1976 

4-2303 

4-2635 

4.2972 

3315 

13 

77 

3315 

3662 

4015 

4373 

4737 

5107 

5483 

5864 

6252 

6646 

7046 

12 

78 

7046 

7453 

7867 

8288 

8716 

9152 

9594 

5.0045 

5-0504 

5.0970 

5.1446 

11 

79 

5.1446 

5"!929 

5.2422 

5.2924 

5-3435 

5-3955 

5-4486 

5026 

5578 

6140 

6713 

1O 

80 

81 
82 
83 

5-67I3 
6.3138 
7-"54 
8.1443 

5-7297 
6-3859 

7.2066 
8.2636 

5-7894 
6.4596 
7.3002 
8.3863 

5.8502 
6-5350 
7.3962 
8.5126 

5-9i24 
6.6122 

7-4947 
8.6427 

I'?758 
6.6912 

7.5958 
8.7769 

6.0405 
6.7720 
7.6996 
8.9152 

6.1066 
6.8548 
7.8062 
9.0579 

6.1742 

6-9395 
7-9158 
9.2052 

6.2432 
7.0264 
8.0285 
9-3572 

6.3138 
7-II54 
8.1443 
9.5144 

9 

7' 
6 

84 

9.5144 

9.6768 

9.8448 

10.0187 

10.1988 

10.3854 

10.5789 

10.7797 

10.9882 

11.2048 

11.4301 

& 

85 

86 

11.4301 
14.3007 

11.6645 
14.6685 

11.9087 
I5-0557 

12.1632 
15.4638 

12.4289 
15.8945 

12.7062 
16.3499 

12.9962 
16.8319 

13.2996 
I7.3432 

13.6174 
17.8863 

I3-9507 
18.4645 

14.3007 
19.0811 

4 

3 

!87 

88 

89 

19.0811 
28.6363 
57.2900 

19.7403 
30.1446 
63.6567 

^0.4465 
31.8205 
71.6151 

21.2049 

33.6935 
81.8470 

22.0217 
35.8006 
954895 

22.9038 
38.1885 
i  14.5886 

23-8593 
40.9174 

24.8978 
44.0661 
190.9842 

26.0307 

47-7395 
286.4777 

27.2715 
52.0807 
572.9572 

28.6363 
57.2900 

00 

2 

1 
0 

Q> 

60 

54' 

48' 

42' 

36' 

30' 

24' 

18' 

12' 

6' 

0' 

S 

1 

1*0 

0°.9 

0°.8 

0°.7 

06 

0°.5 

O°.4 

O°.3 

0°.2 

0*1 

o°.o 

1 

Natural  Cotangents. 

84 


TABLE  XVII. 


Natural  Versed  Sines. 

i 

o°.o 

01 

0  .2 

0°.3 

0°.4 

0^.5 

0.6 

0°.7 

0°.8 

0°.9 

O 

O' 

6' 

12' 

IS' 

24 

30 

36' 

42' 

48'    54' 

Q 

. 

0 

o.oooo 

0.0000 

o.oooo 

0.0000 

o.oooo 

0.0000 

O.OOOI 

O.OOOI 

O.OOOI 

O.OOOI 

1 

OOO2 

OOO2 

OOO2 

0003 

0003 

0003  1  0004 

0004 

0005 

0015  : 

2 

0006 

0007 

0007 

0008 

0009 

0010 

0010 

OOII 

0012 

0003 

3 

0014 

0015 

0016 

0017 

0018 

0019 

OO2O 

OO2I 

OO22 

0013 

4 

0024 

0026 

0027 

0028 

0029 

0031 

0032 

0034 

°°35 

°°37 

5 

0.0038 

0.0040 

0.0041 

0.0043 

0.0044 

0.0046 

0.0048 

0.0049 

O.OO5I 

0.0053 

6 

0055 

0057 

0058 

0060 

0062 

0064 

0066 

0068 

0070 

0072 

7 

0075 

0077 

0079 

0081 

0083 

0086 

0088 

0090 

0093 

0095 

8 

0097 

OIOO 

OIO2 

0105 

0107 

OIIO 

OII2 

0115 

0118 

0120 

9 

0123 

0126 

0129 

0131 

0134 

0137 

0140 

0143 

0146 

0149 

10 

0.0152 

0.0155 

0.0158 

0.0161 

0.0164 

0.0167 

O.OI7I 

0.0174 

0.0177 

O.OlSo 

11 

0184 

0187 

OIOX) 

0194 

0197 

0201 

O2O4 

0208 

021  1 

O2I5 

12 

0219 

O222 

0226 

0230 

0233 

0237 

024! 

0245 

0249 

0252 

13 

0256 

0260 

0264 

0268 

0272 

0276 

0280 

0285 

0289 

0293  I 

14 

0297 

0301 

0306 

0310 

°3  H 

0319 

0323 

0327 

0332 

0336 

15 

0.0341 

0.0345 

0.0350 

0.0354 

0-0359 

0.0364 

0.0368 

0.0373 

0.0378 

0.0383  I 

16 

0387 

0392 

0397 

0402 

0407 

0412 

0417 

0422 

0427 

0432 

17 

0437 

0442 

0447 

0452 

0458 

0463 

0468 

0473 

0479 

0484! 

18 

0489 

0495 

0500 

0506 

0511 

0517 

0522 

0528 

°534 

0539  1 

19 

0545 

0551 

0556 

0562 

0568 

0574 

0579 

0585 

0591 

0597  | 

2O 

0.0603 

0.0609 

0.0615 

0.0621 

0.0627 

0.0633 

0.0639 

0.0646 

0.0652 

0.0658  i 

21 

0664 

0670 

0677 

0683 

0689 

0696 

0702 

0709 

0715 

0722  ! 

22 

0728 

°735 

0741 

0748 

0755 

0761 

0768 

0775 

0788  i 

23 

0795 

0802 

0809 

0816 

0822 

0829 

0836 

0843 

0850 

0857, 

24 

0865 

0872 

0879 

0886 

0893 

0900 

0908 

0915 

0922 

0930  ; 

25 

0.0937 

0.0944 

0.0952 

0.0959 

0.0967 

0.0974 

0.0982 

0.0989 

0.0997 

0.1004 

26 

IOI2 

IO2O 

1027 

1035 

1043 

1051 

1058 

1066 

1074 

1082 

27 

IOOX) 

1098 

1106 

1114 

1  122 

1130 

II38 

1146 

H54 

1162 

28 

II7I 

1179 

1187 

1195 

I2O4 

1212 

1220 

1229 

1237 

1245  ' 

29 

1262 

1271 

1279 

1288 

1296 

1305 

1314 

1322 

1331 

3O 

0.1340 

0.1348 

0.1357 

0.1366 

°-I375 

0.1384 

o.  393 

0.1401 

O.I4IO 

0.1419 

31 

1428 

H37 

1446 

H55 

1464 

H74 

1492 

I5O1 

1510 

32 
33 

1520 
1613 

1529 
1623 

1538 
1632 

1547 
1642 

i|57 
1652 

1566 

1661 

575 
671 

Io8o 

1594 
1690 

1604 
1700 

34 

1710 

1719 

1729 

1739 

J749 

1759 

769 

1779 

1789 

1798 

35 

o.i  808 

O.l8l9 

0.1829 

0.1839 

0.1849 

0.1859 

0.1869 

0.1879 

0.1889 

0.1900 

36 

1910 

1920 

1930 

1941 

195  * 

1961 

1972 

1982 

1993 

2003 

37 

2014 

2O24 

2035 

2045 

2056 

2066 

2077 

2088 

2098 

2109 

38 

2120 

2131 

2141 

2152 

2163 

2174 

2185 

2196 

2207 

2218 

39 

2229 

2240 

2251 

2262 

2273 

2284 

2295 

2306 

2317 

2328 

40 

0.2340 

0.2351 

0.2362 

0-2373 

0.2385 

0.2396 

0.2407 

0.2419 

0.2430 

0.2441 

41 

2453 

2464 

2476 

2487 

2499 

2510 

2522 

2534 

2545 

2557 

42 

2569 

2580 

2592 

2604 

2615 

2627 

2639 

2651 

2663 

2075 

43 

2686 

2698 

2710 

2722 

2734 

2746 

2758 

2770 

2782 

2794 

44 

2807 

2819 

2831 

2843 

2855 

2867 

2880 

2892 

2904 

2917  j 

I  45 

O.2929 

0.2941 

0.2954 

0.2966 

0.2978 

0.2991 

0.3003 

0.3016 

0.3028 

0.3041 

46 

3°53 

3066 

3°79 

3091 

3I04 

3116 

3129 

3142 

3155 

3167 

47 

48 
49 

3I80 
3309 

3439 

3193 
3322 

3453 

3206 

3335 
3466 

3218 
3348 
3479 

3231 
3492 

3244 

3374 
3506 

32^7 
33»7 

3270 
3400 
3532 

3283 
3413 

3545 

3296  ; 
3426 
3559 

50 

0.3572 

0.3586 

0-3599 

0.3612 

0.3626 

0.3639 

0.3653 

0.3666 

0.3680 

0-3693 

51 

3707 

3720 

3734 

3748 

376i 

3775 

3789 

3802 

3816 

3830 

52 

3843 

3857 

3871 

3885 

3899 

3912 

3926 

3940 

3954 

3968 

53 

3982 

3996 

4010 

4024 

4038 

4052 

4066 

4080 

4094 

4108  | 

54 

4122 

4I50 

4165 

4179 

4193 

4207 

4221 

4236 

4250 

55 

0.4264 

0.4279 

0.4293 

0.4307 

04322 

0.4336 

0-435° 

0.4365 

0-4379 

0-4394 

56 
57 

4408 
4554 

4423 
4568 

4437 
4583 

4452 
4598 

4466 
4612 

4627 

4495 
4642 

45  10 
4656 

4524 
4671 

4686  | 

58 

4701 

4716 

473° 

4745 

476o 

4775 

4790 

4805 

4820 

4835 

59 

4850 

4865 

4880 

4895 

4910 

4925 

4940 

4955 

497° 

49% 

i 

TABLE  XVIII. 


85 


Decimal  Parts  and  their  Multiples  of  a  Day. 

Hrs. 

Dec'l 
Parts 
nf 

Multipliers. 
Hrs. 

Dec'l 
Parts 
of 

Multipliers. 

vJl 

Day. 

2  3 

4 

5 

6   7  8  9  ! 

Day. 

»j  3 

4 

5 

6 

7 

8 

9 

0.0 

o.ooo 

o.oo  o.oo 

0.00 

0.00 

o.oo  o.oo  o.oo  o.oo  !  12.O 

0.500 

I.OO 

1.50 

2.OO 

2.50 

3.00 

3-5o 

4.00 

4-5° 

2 

008 

02  03 

03 

04 

05!  06 

07  08    2 

508 

02 

53 

°3 

54 

°5 

56 

07 

58 

4 

017 

03  05 

07 

08 

IO 

12 

13  15    4 

517 

°3 

55 

07 

58 

10 

62 

13 

65, 

6 

025 

05  08 

IO 

13 

15 

18  20j  23    6 

525 

05  5* 

10 

63 

15 

68 

20 

73 

8 

033 

07J  10 

13 

17 

20 

23  27  30    8 

533 

07  60 

13 

67 

20 

73 

27 

1.0 

0.042 

0.080.13 

0.17 

0.21 

0.25 

0.29 

0.330.38  13.O 

0.542 

i.  08 

1.63 

2.17 

2.71 

3-25 

3-79 

4-33 

4.88 

2 

050 

Ioi  *5 

20 

25 

3° 

35 

4°  45    2 

550 

10 

65   20 

75 

3° 

85 

40 

95 

4 

058 

12 

18 

23 

29 

35 

41 

47  53  |   4 

558 

12 

68  23 

79 

35 

47 

5-03 

6 

067 

13 

20 

27 

33 

40 

47 

53  60    6 

567 

'3 

7o|  27 

83 

40 

97 

53 

10 

8 

075 

15 

23 

30 

37 

45 

53 

60  68    8 

575 

15 

73 

3° 

87 

45 

4-03 

60 

18 

2.0 

0.083 

0.170.25 

0-33 

0.42 

0.50 

0.58 

0.670.75'!  14.0 

0-583 

1.17 

1.75 

2-33 

2.92 

3-5° 

4.08 

4.67 

5-25 

2 

092 

18 

28 

37 

46 

55 

64 

73  83 

2 

592 

18 

78 

37 

96 

55 

H 

73 

33 

4 

IOO 

20 

3° 

40 

50 

60 

70 

80  90 

4 

600 

20 

80 

40 

3.00 

60 

20 

80 

40 

6 

108 

22 

'33 

43 

65 

76 

87!  98 

6 

608 

22 

83 

43 

04 

65 

26 

87 

48 

8 

117 

23 

35 

47 

58 

70 

82 

93  J-°5 

8 

617 

23 

85 

47 

08 

70 

32 

93 

55 

3.0 

0.125 

0.25 

0.38 

0.50 

0.63 

0-75 

0.88 

1.001.13 

15.O 

0.625 

1.25 

1.88 

2.50 

3-13 

3-75 

4-38 

5.00 

5.63 

2 

27 

40 

53 

67 

80 

93 

07 

20 

2 

633 

27 

90 

53 

17 

80 

43 

07 

70 

4 

142 

28 

43 

57 

71 

85 

99 

13 

28 

4 

642 

28 

93 

57 

21 

85 

49 

13 

78 

6 

150 

3° 

45 

60 

75 

90 

1.05 

201  35 

6 

650 

3° 

95 

60 

25 

90 

55 

20 

85 

8 

158 

32 

48 

63 

79 

95 

ii 

27  43 

8 

658 

32 

98 

63 

29 

95 

61 

27 

93 

4.0 

0.167 

o-33 

0.50 

0.67 

0.83 

I.OO 

1.17 

1.33  1.50 

16.0 

0.667 

i-33 

2.00 

2.67 

3-33 

4.00 

4.67 

5-33 

6.00 

2 

175 

35 

53 

70 

87 

05 

23  40|  58 

2 

675 

35 

03 

37 

05 

73 

40 

08 

4 

37 

55 

73 

91 

10  28]  47!  65 

4 

683 

37 

05 

73 

41 

10 

78 

47 

15 

6 

192 

38 

77 

95 

15  34  53 

73 

6 

692 

38 

08 

77 

45 

15 

84 

53 

23 

8 

200 

40 

6O 

80 

l-OO 

20  40  60 

80 

8 

700 

40 

10 

80 

50 

20 

90 

00 

3° 

5.0 

0.208 

0.42 

0.63 

0.83 

1.04 

1.251.461.67 

I.881J17.0 

0.708 

1.42 

2.13 

2.83 

3-54 

4-25 

4.96 

5.67 

6.38 

2 

217 

43 

65 

87 

08 

30   52 

73 

95    2 

717 

43 

15 

87 

I8 

30 

5.02 

73 

45 

4 

225 

45 

68 

90 

12 

35 

58 

80 

2.03!    4 

725 

45 

18 

90 

62 

35 

08 

80 

53 

6 

233 

47 

70 

93 

17 

40 

63 

87 

10    6 

733 

47 

20 

93 

67 

40 

13 

87 

00 

8 

242 

48 

73 

97 

21 

45 

69 

93 

i8||   8 

742 

48 

23 

97 

7i 

45 

19 

93 

68 

6.0 

0.250 

0.50 

0-75 

I.OO 

1.25 

1.50 

i-75 

2.00 

2.25  18.0 

0.750 

1.50 

2.25 

3-oo 

3-75 

4-50 

5-25 

6.00 

6-75 

2 

258 

52 

78 

°3 

29 

55 

81 

07 

331   2 

758 

S2 

28 

03 

79 

55 

31 

07 

83 

4 

267 

53 

80 

07 

33 

60 

87 

13 

40 

4 

767 

53 

30 

07 

83 

60 

37 

90 

6 

275 

55 

83 

10 

37 

65 

93 

20 

48 

6 

775 

55 

33 

10 

87 

65 

43 

20 

98 

8 

283 

57 

85 

13 

42 

70 

98 

27 

55 

8 

783 

57 

35 

13 

92 

70 

48 

27 

7-05 

ro 

0.292 

0.58 

0.88 

1.17 

1.46 

i-75 

2.04 

2-33 

2.63 

190 

0.792 

1.58 

2.38 

3-'7 

3-96 

4-75 

5-54 

6-33 

7-13 

2 

300 

60 

90 

20 

50 

80 

10 

40 

70 

2 

800 

60 

40 

20 

4.00 

80 

60 

40 

20 

4 

308 

62 

93 

23 

54 

85 

16 

47 

78 

4 

808 

62 

43 

23 

04 

85 

66 

47 

28 

6 

63 

95 

27 

58 

90 

22 

53 

85 

6 

817 

63 

45 

27 

08 

90 

72 

53 

35 

8 

325 

65 

98 

30 

62 

95 

28 

60 

93 

8 

825 

65 

48 

30 

12 

95 

78 

00 

43 

8.0 

0-333 

0.67 

I.OO 

i-33 

1.67 

2.00 

2-33 

2.67 

3-oo 

20.0 

0-833 

1.67 

2.50 

3-33 

4.17 

5.00 

5-83 

6.67 

7-50 

2 

342 

68 

03 

37 

71 

05 

39 

73  08 

2 

842 

68 

53 

37 

21 

05  89 

73 

58 

4 

350 

7o 

°5 

40 

75 

10 

45 

80 

15  i   4 

850 

70 

55 

40 

25 

io|  95 

80 

65! 

6 

358 

72 

08 

43 

79 

15 

87 

23    6 

858 

72 

58 

43 

29 

15 

6.01 

87 

73 

8 

367 

73 

IO 

47 

83 

20 

57 

93 

3°    8 

867 

73 

60 

47 

33 

20 

07 

93 

80 

9.0 

0.375 

°-75 

1.13 

1.50 

1.87 

2.25 

2-63 

3.00 

3.38  21.0 

0.875 

i-75 

2.63 

3-50 

4-37 

5-25 

6.13 

7.00 

7.88 

2 

383 

77 

I  ; 

53 

92 

30 

68 

07 

45 

2 

883 

77 

65 

53 

42 

3° 

18 

07 

95 

4 

6 

392 
400 

78 
80 

20 

57 
60 

96 

2.OO 

35 
40 

80 

13 

20 

8 

4 

6 

892 
900 

i 

68 
70 

46 
50 

35 
4° 

24 
3° 

13 

20 

8.03 

10 

8 

408 

82 

23 

63 

04 

45 

86 

27 

68    8 

908 

82 

73 

63 

54 

45 

36 

27 

18 

10.0 

0.417 

0.83 

1.25 

1.67 

2.08 

2.50 

2.92 

3-33 

3-75 

22.0 

0.917 

1.83 

2.75 

3-67 

4.58 

5-50 

6.42 

7-33 

8.25 

2 

425 

85 

28 

70 

12 

55 

98 

40 

83 

2 

925 

85 

78 

70 

62 

55 

48 

40 

33 

4 

433 

87 

3° 

73 

17 

60 

3-°3 

47 

90 

4 

933 

87 

80 

73 

67 

60 

53 

47 

40 

6 

442 

88 

33 

77 

21 

65 

09 

53 

98 

6 

942 

88 

f3 

V 

71 

65 

59 

53 

48 

8 

45° 

90 

35 

80 

25 

70 

15 

60 

4-05 

8 

95° 

90 

85 

80 

75 

70 

65 

60 

55 

110 

0.458 

0.92 

1.38 

1.83 

2.29 

2-75 

3-21 

3-67 

4-13 

23.0 

0.958 

1.92 

2.88 

3-83 

4-79 

5-75 

6.71 

7-67 

8.63 

2 

467 

93 

40 

87 

33 

80 

27 

73 

20 

2 

967 

93 

90 

87 

83 

80 

77 

73 

70 

4 

475 

95 

43 

90 

37 

85 

33 

80 

28    4 

975 

95 

93 

90 

87 

85 

80 

78 

6 

483 

97 

45 

93 

42 

90 

38 

87 

35    6 

983 

97 

95 

93 

92 

90 

88 

87 

85 

8 

492 

98 

47 

97 

46 

95 

44 

93 

43!    8 

992 

98 

97 

97 

96 

95 

94 

93 

93 

12.O  0.500 

I.OO 

1.50 

2.OO 

2.50 

3-oo 

3-5° 

4.004.50  24.O 

I.OOO 

2.OO 

3.00 

4.00 

5.00 

Ob 

7.00 

S.oo 

9.00! 

1 

86 


TABLE  XIX. 


1 

Decimal  Equivalents  to  two  places  of  Common  Fractions. 

Numerator  of  Fraction. 

• 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

D. 

10 

00 

10 

20 

3° 

40 

5° 

60 

70 

80 

90 

" 

10 

11 

oo 

09 

18 

27 

36 

45 

54 

64 

73 

82 

11 

12 

00 

08 

17 

25 

33 

42 

5° 

58 

67 

75 

83 

91 

12 

13 

00 

08 

15 

23 

31 

38 

46 

54 

61 

69 

77 

85 

92 

13 

14 

oo 

07 

H 

21 

29 

36 

43 

50 

57 

64 

7i 

78 

86 

93 

14 

15 

00 

07 

'3 

20 

27 

33 

40 

47 

53 

60 

67 

73 

80 

87 

93 

15 

16 

oo 

06 

12 

19 

25 

31 

37 

44 

50 

56 

62 

69 

75 

Si 

87 

94 

16 

17 

00 

06 

12 

18 

23 

35 

41 

47 

53 

59 

*5 

7i 

76 

82 

88 

94 

17 

18 

oo 

06 

II 

17 

22 

28 

33 

39 

44 

5° 

56 

61 

67 

72 

78 

83 

89 

94 

18 

19 

oo 

05 

IO 

16 

21 

26 

32 

37 

42 

47 

53 

58 

63 

68 

74 

79 

84 

89 

95 

19 

2O 

00 

05 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

60 

65 

70 

75 

80 

SoS 

90 

95 

2O 

21 

oo 

05 

09 

14 

IQ 

24 

29 

33 

38 

43 

48 

52 

57 

62 

67 

7i 

76 

81 

86 

90 

95 

21 

22 

00 

04 

09 

14 

18 

23 

27 

32 

36 

4i 

45 

5° 

54 

59 

64 

68 

73 

77 

82 

86 

9i 

22 

23 

oo 

04 

09 

*3 

*7 

22 

26 

3° 

35 

39 

43 

48 

52 

56 

61 

65 

70 

74 

78 

83 

87 

23 

24 

00 

04 

08 

12 

17 

21 

25 

29 

33 

37 

42 

46 

So 

54 

58 

62 

67 

7i 

75 

79 

83 

24 

25 

oo 

04 

08 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

56 

60 

64 

68 

72 

76 

80 

25 

26 

00 

04 

08 

II 

15 

19 

23 

27 

31 

35 

38 

43 

46 

50 

54 

58 

61 

65 

69 

73 

77 

26 

27 

00 

04 

07 

II 

15 

18 

22 

26 

3° 

33 

37 

4i 

44 

48 

52 

56 

59 

63 

67 

70 

74 

27 

28 

oo 

04 

07 

II 

14 

18 

21 

25 

29 

32 

36 

39 

43 

46 

50 

54 

57 

61 

64 

68 

7i 

28 

29 

00 

°3 

07 

10 

14 

!7 

21 

24 

28 

31 

34 

38 

4i 

45 

48 

52 

55 

59 

62 

65 

69 

29 

3O 
31 

oo 

00 

03 

°3 

3 

IO 
10 

13 

!3 

;i 

20 

19 

23 
23 

27 
26 

30 

33 

32 

37 
35 

40 
39 

43 
42 

47 
45 

5° 
48 

53 

52 

57 

55 

60 

58 

63 
61 

67 
64 

30 
31 

32 

00 

03 

06 

09 

12 

16 

19 

22 

25 

28 

3* 

34 

37 

4i 

44 

47 

50 

53 

56 

59 

62 

32 

33 

oo 

03 

06 

09 

12 

15 

18 

21 

24 

27 

3<> 

33 

36 

39 

42 

45 

48 

5i 

54 

5f 

61 

33 

34 

oo 

°3 

06 

09 

12 

15 

18 

21 

23 

26 

29 

32 

35 

38 

4i 

44 

47 

50 

53 

56 

59 

34 

35 

oo 

03 

06 

09 

II 

14 

!7 

20 

23 

26 

29 

31 

34 

37 

40 

43 

46 

49 

51 

54 

57 

35 

36 

00 

°3 

06 

08 

II 

14 

17 

19 

22 

25 

28 

31 

33 

36 

39 

42 

44 

47 

5° 

53 

56 

36 

37 

00 

03 

05 

08 

II 

13 

16 

19 

22 

24 

27 

30 

32 

35 

38 

40 

43 

46 

49 

51 

54 

37 

38 

oo 

03 

05 

08 

IO 

J3 

16 

18 

21 

24 

26 

29 

32 

34 

37 

39 

42 

45 

47 

50 

53 

38 

39 

oo 

°3 

05 

08 

10 

13 

15 

18 

2O 

23 

26 

28 

3i 

33 

36 

38 

4i 

43 

46 

49 

51 

39 

4O 

oo 

02 

°5 

07 

IO 

12 

15 

17 

20 

22 

25 

27 

30 

32 

35 

37 

40 

42 

45 

47 

50 

40 

42 

oo 

02 

°5 

07 

09 

12 

14 

17 

19 

21 

24 

26 

29 

3i 

33 

36 

38 

40 

43 

45 

48 

42 

44 

00 

02 

04 

06 

09 

II 

14 

16 

18 

2O 

23 

25 

27 

29 

32 

34 

36 

39 

4i 

43 

45 

44 

46 

oo 

02 

04 

06 

09 

II 

13 

15 

17 

20 

22 

24 

26 

28 

30 

33 

35 

37 

39 

4i 

43 

46 

48 

oo 

02 

04 

06 

08 

IO 

12 

15 

17 

19 

21 

23 

25 

27 

29 

3» 

33 

35 

37 

39 

42 

48 

1  50 

00 

02 

04 

06 

08 

10 

12 

14 

16 

18 

2O 

22 

24 

26 

28 

3° 

32 

34 

36 

38 

40 

50 

I  52 

oo 

02 

04 

06 

08 

IO 

II 

13 

15 

*  7 

19 

21 

23 

25 

27 

20 

31 

33 

35 

36 

38 

52 

!  54 

00 

02 

04 

06 

07 

09 

II 

*3 

15 

17 

18 

2O 

22 

24 

26 

2o 

30 

3i 

33 

35 

37 

54 

56 

00 

02 

04 

05 

07 

09 

II 

12 

H 

16 

18 

20 

21 

23 

25 

27 

29 

3° 

32 

34 

36 

56 

58 

oo 

02 

°3 

05 

07 

09 

10 

12 

14 

15 

17 

19 

21 

22 

24 

26 

27 

29 

31 

33 

34 

58 

60 

00 

02 

03 

°5 

07 

08 

IO 

12 

'3 

15 

17 

18 

2O 

22 

23 

25 

27 

28 

3° 

32 

33 

60 

62 

oo 

02 

03 

05 

06 

08 

10 

II 

13 

14 

16 

18 

19 

21 

23 

24 

26 

27 

29 

3i 

32 

62 

64 

oo 

02 

°3 

05 

06 

08 

09 

II 

12 

i.: 

16 

17 

18 

20 

22 

23 

25 

26 

28 

3° 

31 

64 

66 

oo 

OI 

°3 

°5 

06 

08 

09 

II 

12 

A 

15 

17 

18 

2O 

21 

23 

24 

26 

27 

3° 

66 

68 

00 

01 

°3 

04 

06 

07 

09 

10 

12 

2 

15 

16 

18 

19 

21 

22 

23 

25 

26  i  28 

29 

68 

7O 

oo 

01 

°3 

04 

06 

07 

09 

IO 

II 

2 

14 

16 

17 

JO 

2O 

21 

23 

24 

26  27 

29 

7O 

72 

00 

01 

°3 

04 

06 

07 

08 

10 

II 

2 

H 

15 

17 

18 

19 

21 

22   24 

25  !  26 

28 

72 

74 

00 

01 

°3 

04 

05 

07 

08 

09 

II 

2 

*3 

15 

16 

18 

19 

20 

22 

23 

24  26 

27 

74 

76 

oo 

OI 

°3 

04 

°i 

07 

08 

09 

10 

2 

13 

H 

16 

17 

18 

2O 

21 

22 

24 

25 

26 

76 

78 

00 

OI 

03 

04 

Of 

06 

08 

09 

IO 

II 

*3 

14 

15 

»7 

18 

19 

2O 

22 

23 

24 

26 

78 

80 

oo 

01 

02 

04 

05 

06 

07 

09 

10 

II 

12 

14 

15 

16 

17 

19 

20 

21 

22 

24 

25 

8O 

82 

oo 

OI 

02 

04 

°! 

06 

07 

08 

IO 

II 

12 

*3 

15 

16 

17 

18 

19 

21 

22 

23 

24 

N2 

84 

oo 

01 

02 

04 

°l 

06 

07 

08 

09 

II 

12 

!3 

14 

15 

17 

18 

19   20 

21 

23 

24 

84 

86 

00 

01 

02 

°3 

05 

06 

07 

08 

09 

IO 

12 

!3 

H 

15 

16 

17 

19   20 

21 

22 

23 

86 

88 

oo 

01 

02 

03 

04 

06 

07 

08 

09 

10 

II 

12 

H 

15 

16 

'7 

18 

19 

2O 

22 

23 

88 

9O 

oo 

01 

02 

°3 

04 

06 

07 

08 

09 

10 

II 

12 

13 

'4 

15 

17 

18 

19 

20 

21 

22 

90 

92 

oo 

01 

02 

°3 

ot 

05 

06 

08 

09 

IO 

11 

12 

*3 

'4 

15 

16 

17 

18 

20 

21 

22 

92 

94 

00 

01 

02 

°3 

Qi 

Of 

06 

07 

08 

09 

II 

12 

13 

H 

15 

16 

17 

18 

19 

20 

21 

94 

96 

00 

OI 

02 

03 

Ot 

Of 

06 

07 

08 

09 

10 

II 

12 

J3 

15 

16 

17 

18 

19 

20 

21 

96 

98 

oo 

OI 

02 

03 

Ot 

Of 

06 

07 

08 

oc 

10 

II 

12 

13 

H 

15 

16 

17 

18 

19 

20 

98 

100 

00 

01 

02(03 

ot 

05 

06 

07 

08 

09 

10 

II 

12 

13 

H 

15 

16 

17 

18 

19 

2O 

100 

TABLE  XIX. 


87 


Decimal  Equivalents  to  two  places  of  Common  Fractions. 

Numerator  of  Fraction. 

D. 

1       1 

i 

D. 

20 

21 

22 

23 

21 

25 

26 

2728 

29 

30 

31  32  33 

31 

35 

36 

37 

3§ 

39 

10 

20 

20 

21 

9S 

21  ! 

22 

9S 

i 

22 

i    23 

87 

91 

95 

1 

23 

21 

83 

87 

92 

96 

21 

25 

83 

84 

88 

92    96 

25  i 

26 

77 

81 

8S 

88    92 

96 

26 

|    27 

74 

78 

81 

8S 

89 

9} 

95 

27 

2§ 

7S 

78 

82 

85 

89 

93 

96 

1 

28 

29 

6} 

72 

79    83 

83 

93 

93 

97 

29 

30 

67 

70 

73    77 

83 

8} 

87 

90 

93 

97 

i 

30 

31 

64 

68 

74 

77 

Si 

84 

87 

90 

94 

97 

31 

32 

62 

65 

69 

72 

7S 

78 

Si    84 

87 

9i 

94 

97 

[              ' 

32 

33 

or 

64    67 

70 

79    82 

SS 

88 

91 

94197 

33 

31 

S) 

62 

65    68 

71 

73 

79 

82 

85 

88 

91    94 

97 

31 

35 

57 

63 

61    65 

69 

7i 

74 

77 

83 

83 

85 

89    91 

94 

97 

35 

36 

5*1  61 

64 

67 

6? 

72  1  75 

78 

81 

83 

86    89 

92 

94 

97 

36 

37 

S4 

57 

59 

62 

6S 

68 

7o    73 

76 

78 

Si 

84   86 

89 

92 

95 

97 

37 

3§ 

S3 

SS 

58    60 

63 

65 

68    71 

74 

76 

79 

81    84 

87    89 

92 

95 

97 

38 

39 

51 

54 

5^    59 

61 

64 

67   69 

72 

74 

77 

79    82 

85     87 

90 

92 

95!  97 

39 

10 

S3 

S2 

SS 

S7 

60 

62 

6S    67 

70 

72 

7S 

77   80 

82!  85 

87 

90  |  92  i  95 

97 

10 

U 

4> 

52 

5S 

57 

59 

62    64 

67 

69 

74    76 

78  j  81 

86 

88  [  90 

93 

9S 

12 

11 

45 

4^ 

5^ 

SS 

57 

59    61 

64 

65 

68 

70!  73 

7S 

77 

83 

82 

84 

So 

89 

91 

11 

16 

43 

43 

52 

54 

55   59 

61 

63 

6S 

67(69 

72 

74 

76 

78. 

So 

83 

8S 

87 

16 

18 

42 

44 

45 

48 

53 

52 

54   55 

58 

60 

62 

65 

67 

69 

73 

75 

77 

79 

Si 

83 

18 

50 

40 

42 

44 

45 

48 

50 

S2 

54 

S6 

58 

60 

62 

64 

65 

68 

70 

72 

74 

76 

78 

80 

50 

5^ 

33 

43 

42 

44 

43 

48 

50 

S4 

55 

S8 

60 

61 

63 

6S 

67 

69 

7i 

73 

75 

77 

52 

51 

37 

39 

41 

43 

44 

45 

48    So 

52 

54 

56 

S7 

59 

61 

63 

6S 

67 

69 

7° 

72 

74 

51 

56 

30 

37 

39  |4i 

43 

4S 

46J48 

So 

52 

S4 

SS 

57 

59 

61 

62 

64 

65 

68 

56 

5& 

34 

38  i  43i  4l 

43 

45    47 

48 

53 

55 

57 

59 

60 

62 

64 

65 

67 

69 

58 

69 

33 

35 

37    38 

43 

42 

43   45 

47 

48 

5° 

S2 

S3 

55    57 

S8 

60 

62 

63 

65 

6/ 

60  ! 

62 

32 

34 

35137 

39 

40 

42i  44 

4) 

47 

48 

50 

52 

53    55 

56 

S8 

60 

61 

63 

64 

62  I 

61 

31 

33 

34 

33 

37 

39 

41    42 

44 

4S 

47 

48!  50 

52 

S3 

55 

56 

58 

59 

61 

62 

61 

66 

33 

32 

33 

35 

38 

31   4i 

42 

44 

4S 

47 

48 

So 

52 

S3 

SS 

56 

ss 

S9 

61 

66 

63 

29 

3° 

32    34 

35 

37 

38(40 

4i 

43 

44 

46 

47 

48 

5° 

51 

53 

54 

56 

57 

59 

68 

70 

29 

33 

3i    33 

34 

35 

37 

39 

40 

41 

43 

44 

46 

47 

48 

5o 

5i 

53 

54 

56 

57 

70  1 

72 

28 

29 

31 

32 

33 

35 

37 

39 

40 

42 

43 

44 

46 

47 

49 

5° 

5i 

53 

S4 

56 

72 

71 

27 

28 

33 

31 

32 

34 

35 

36 

38 

39 

40 

42 

43 

45 

46 

47 

49 

Si 

S3 

54 

71 

76 

23 

28 

29   30 

32 

33 

34 

35 

37 

38 

39 

41 

42 

43 

4S 

46 

47 

49 

5o 

Si 

53 

76 

78 

23 

27 

28    29 

31 

32 

33 

35 

36 

37 

38 

40 

41 

42 

44 

45 

46 

47 

49 

78j 

80 

25 

23 

27   29 

30 

31 

32 

34 

35 

36 

37 

39 

40 

41 

42 

44 

45 

46 

47 

49 

So 

80; 

82 

24 

23 

27 

28 

29 

3° 

32 

33 

34 

35 

37 

38 

39 

40 

43 

44 

45 

46 

48 

49 

82 

81 

24 

25 

2t> 

27 

29 

30 

32 

33 

34 

36 

37 

38 

39 

40 

42 

43 

44 

45 

46 

48 

81 

86 

23 

24 

25 

27 

28 

29 

3° 

3i 

32 

34 

35 

3° 

37 

38 

39 

41 

42 

43 

44 

4S 

46 

86 

88 

23 

24 

25 

26 

27 

28 

23 

31 

32 

33 

34 

35 

36 

37 

39 

40 

41 

42 

43 

44 

45 

88 

90 

22 

23 

24 

25 

27 

28 

29 

30 

31 

32 

33 

34 

36 

37 

38 

39 

40 

41 

42 

43 

44 

90 

92 

22 

23 

24 

25 

23 

27 

28 

29 

30 

31 

33 

34 

35 

36 

37 

38 

39 

40 

42 

43 

92 

91 

21 

22 

23 

24 

25 

27 

28 

29 

3° 

31 

32 

33 

34 

35 

36 

37 

38 

39 

40 

41 

42 

91 

93 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

31 

32 

33 

34 

35 

36 

37 

38 

39 

41 

42 

96 

98 

20 

21 

22 

23 

24 

25 

23 

27 

29 

3° 

32 

33 

34 

35 

36 

37 

38 

39 

40 

98 

l.OO 

2O 

21 

22 

23 

24 

25 

26 

27 

28 

29 

3° 

3i 

32 

33 

34 

35 

36 

37 

38 

39 

40 

100 

». 

1 

5 

10 

15 

20 

25 

3} 

33 

40  45 

59 

55 

60 

65 

70 

75 

80 

85 

90 

95 

103 

D. 

125 

OI 

04 

08 

12 

16 

20 

24 

28 

32 

36 

40 

44)48 

52 

56 

60 

64 

68 

72 

76 

So 

125! 

J176 

OI  1 

03 

06  08 

ii 

14 

17 

20 

23 

2S 

28 

34 

37 

40 

43 

46 

48 

Si 

54 

57 

1761 

3O1 

00  j 

02 

03 

°5 

07 

08 

10 

12 

13 

15 

17 

18 

20 

22 

23 

25 

26 

28 

30 

32 

33 

301 

D. 

100, 

105 

110113 

120 

123 

(30 

135 

140  143 

150 

155 

160 

163  170 

175 

209 

225 

250275 

300 

D. 

125 

So 

84 

88  i  92    96 

I 

125 

176 

57| 

60 

63165 

68 

71 

74 

77 

80 

82 

8S 

88 

91 

94 

97 

176 

3O1J 

33 

35 

36   38 

40 

4i 

43    45j46 

48 

50 

52    53 

55 

56 

58 

66 

74 

83 

9i 

301 

88 


TABLE  XIX. 


Decimal  Equivalents  to  two  places  of  Common  Fractions. 

Numerator  of  Fraction. 

• 

40 

41 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

• 

42 

95 

97 

42 

44 

93 

95 

98 

44 

46 

8? 

90 

9i 

93 

96 

98 

46 

48 

83 

85 

87 

89 

92 

94 

96 

98 

48  . 

50 

80 

82 

84 

86 

88 

90 

92 

94 

96 

98 

50 

52 

77 

79 

81 

83 

85 

86 

88 

9° 

92 

94 

96 

98 

52 

54 

74 

76 

78 

8b 

81 

83 

85 

87 

89 

9i 

93 

94 

96 

98 

54 

56 

7i 

73 

75 

77 

78 

80 

82 

84 

86 

87 

89 

9i 

93 

95 

96 

98 

56 

58 

69 

7i 

72 

74 

76 

78 

79 

81 

83 

84 

86 

88 

90 

9i 

93 

95 

97   98 

58 

60 

67 

68 

70 

72 

73 

75 

77 

78 

80 

82 

83 

85 

87 

88 

90 

Q2 

93 

95 

97 

98 

6O 

62 

64 

66 

68 

69 

7i 

73 

74 

76 

77 

79 

Si 

82 

84 

85 

87 

89 

90 

92 

94 

95 

97 

62 

64 

62 

64 

66 

67 

69 

70 

72 

73 

75 

77 

78 

80 

81 

83 

84 

86 

87 

89 

9i 

92 

94 

64 

66 

61 

62 

64 

65 

67 

68 

70 

7i 

73 

74 

76 

77 

79 

80 

82 

83 

85 

86 

88 

89 

9i 

66 

68 

59 

63 

62 

63 

65 

66 

68 

69 

7i 

72 

73 

75 

76 

78 

79 

81 

82 

84 

85 

87 

88 

68 

70 

57 

58 

60 

61 

63 

64 

66 

67 

69 

70 

7i 

73 

74 

76 

77 

78 

80 

81 

83 

84 

86 

70 

72 

56 

57 

58 

60 

61 

62 

64 

65 

67 

68 

69 

7i 

72 

74 

75 

76 

78 

79 

81 

82 

83 

72 

74 

54 

55 

57 

58 

59 

61 

62 

63 

65 

66 

68 

69 

70 

72 

73 

74 

76 

77 

78 

80 

81 

74 

76 

53 

54 

55 

56 

58 

59 

60 

62 

63 

64 

65 

67 

68 

70 

7i 

72 

74 

75 

76 

78 

79 

76 

78 

5i 

53 

54 

55 

56 

58 

59 

60 

61 

63 

64 

65 

67 

68 

69 

70 

72 

73 

74 

76 

77 

78 

80 

5° 

51 

52 

54 

55 

56 

57 

59 

60 

61 

62 

64 

65 

66 

67 

69 

70 

7i 

72 

74 

75 

80 

82 

49 

So 

51 

52 

54 

55 

56 

57 

59 

60 

61 

62 

63 

65 

66 

67 

68 

69 

7° 

72 

73 

82  i 

84 

48 

49 

5° 

51 

52 

53 

55 

56 

57 

58 

59 

61 

62 

63 

64 

65 

67 

68 

69 

70 

71 

84 

86 

46 

48 

4-Q 

5° 

51 

52 

53 

55 

56 

57 

58 

59 

60 

61 

63 

64 

65 

66 

67 

68 

70 

86; 

88 

45 

46 

48 

49 

50 

5i 

52 

53 

55 

S^ 

57 

58 

59 

60 

61 

62 

64 

65 

66 

67 

68 

88! 

90 

44- 

45 

47 

48 

49 

5o 

51 

52 

53 

54 

S^ 

57 

58 

59 

60 

61 

62 

63 

64 

66 

67 

90; 

92 

43 

44 

46 

47 

48 

42 

5o 

51 

52 

53 

54 

55 

56 

58 

59 

60 

61 

62 

63 

64 

65 

92  1 

94 

42 

44 

45 

46 

47 

48 

49 

5° 

5i 

52 

53 

54 

55 

56 

57 

58 

60 

61 

62 

63 

64 

94  | 

96 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

CO 

60 

61 

62 

96  j 

98 

4i 

42 

43 

44 

45 

46 

47 

48 

49 

50 

51 

52 

53 

54 

55 

56 

57 

58 

59 

60 

61 

98 

100 

40 

4i 

42 

43 

44 

45 

46 

47 

48 

49 

50 

5i 

52 

53 

54 

55 

S^ 

57 

58 

59 

60 

100 

D. 

63 

61 

62 

iM 

>» 

oo 

j.» 

61 

63 

69 

70 

n 

72 

73 

74 

75 

76 

77 

78 

79 

80 

D. 

62 

97 

98 

62! 

04 

9+ 

95 

97 

9? 

64) 

66 

9i 

92 

94 

95 

97 

98 

66 

68 

83 

93 

9i 

93 

94 

95 

97 

98 

681 

70 

85 

87 

89 

93 

91 

93 

94 

96 

97 

93 

70 

72 

83 

85 

86 

83 

89 

90 

92 

93 

94 

96 

97 

99 

72 

74 

81 

82 

84 

85 

86 

88 

89 

92 

93 

95 

96 

97 

99 

74  i 

76 

79 

SD 

81 

83 

84 

85 

87 

88 

89 

9i 

92 

93 

95 

96 

97 

99 

76! 

78 

77 

78 

79 

81 

82 

83 

85 

86 

87 

88 

90 

9i 

92 

93 

95 

96 

97 

99 

78 

80 

75 

76 

77 

79 

80 

81 

82 

84 

85 

86 

87 

89 

90 

91 

92 

94 

95 

96 

97 

99 

80 

82 

73 

74 

75 

77 

78 

79 

80 

82 

83 

84 

85 

87 

88 

89 

90 

9i 

93 

94 

95 

96 

98 

82 

84 

7i 

72 

74 

75 

76 

77 

78 

80 

81 

82 

83 

84 

86 

87 

88 

89 

90 

92 

93 

94 

95 

84 

86 

70 

7i 

72 

73 

74 

75 

77 

78 

79 

80 

81 

83 

84 

85 

86 

87 

88 

89 

9i 

92 

93 

86! 

!    88 

68 

69 

70 

72 

73 

74 

75 

76 

77 

78 

80 

81 

82 

83 

84 

85 

86 

87 

89 

90 

9i 

88 

90 

67 

68 

69 

70 

7i 

72 

73 

74 

76 

77 

78 

79 

80 

81 

82 

83 

84 

86 

87 

88 

89 

9O 

:    92 

65 

65 

67 

63 

69 

7i 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

83 

84 

85 

86 

87 

92 

91 

64 

65 

66 

67 

68 

69 

70 

71 

72 

73 

74 

76 

77 

78 

79 

80 

81 

82 

83 

84 

85 

94 

96 

62 

63 

65 

66 

67 

68 

69 

70 

7i 

72 

73 

74 

75 

76 

77 

78 

79 

80 

81 

82 

83 

96 

93 

61 

62 

63 

64 

65 

66 

67 

68 

69 

70 

7i 

73 

73 

74 

75 

76 

77 

78 

80 

81 

82 

98 

199 

60 

61 

62 

1L 

JH 

65 

66 

Jli 

68 

69 

TO 

7i 

Zi 

Zi 

74 

21 

21 

2L 

21 

79 

80 

1OO 

D 

1*9 

Si 

"sT 

ii 

81 

83 

"8? 

81 

83"[~89 

91) 

91 

92 

93 

91 

95 

9» 

97 

98 

99 

100 

D. 

82 

98 

99 

82 

81 

95 

96 

98 

99 

84 

86 

93 

94 

95 

96 

98 

99 

86 

88 

9i 

92 

93 

94 

95 

97 

98 

99 

88 

90 

89 

90 

9i 

92 

93 

94 

96 

97 

98 

99 

90 

92 

87 

88 

89 

90 

91 

92 

93 

95 

96 

97 

98 

99 

92 

91 

85 

86 

87 

88 

89 

90 

91 

92 

94 

95 

96 

97 

98 

99 

94 

96 

83 

84 

85 

85 

87 

88 

89 

92 

93 

94 

95 

96 

97 

98 

99 

96 

94 

82 

83 

84 

85    86 

87 

88 

89   90 

9i 

92 

93 

94 

95 

96 

97 

98 

99 

98 

100 

80 

81    82 

83    84 

85 

85 

87 

88 

89 

90 

9i 

92 

93 

94 

95 

96 

97 

98 

99 

100 

TABLE  XX. 


89 


Proportional  Parts. 

I 

Tenths  and  their  Multiples. 

1 

Tenths  and  their  Multiples. 

1 

01 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

j 

O.I 

0.2 

0.3 

0.4 

0.5 

0.6 

O.7 

0.8 

0.9 

1 

O.I 

0.2 

0-3 

0.4 

o-5 

0.6 

0.7 

0.8 

0.9 

56 

5-6 

1  1.2 

16.8 

22.4 

28.0 

33-6 

39-2 

44.8 

504 

2 

O.2 

0.4 

o!6 

0.8 

I.O 

1.2 

1.4 

1.6 

1.8 

57 

5-7 

II.4 

17.1 

22.8 

28.5 

34-2 

39-9 

45-6 

5*'3  ! 

3 

°-3 

0,6 

0.9 

1.2 

1.5 

1.8 

2.1 

2.4 

2.7 

58 

5-8 

11.6 

17.4 

23.2 

29.0 

34-8 

40.6 

46.4 

52.2; 

4 

0.4 

0.8 

1.2 

1.6 

2.O 

2.4 

2.8 

3-2 

3-6 

59 

5-9 

11.8 

17.7 

23-6 

29-5 

354 

41-3 

47-2 

53.1  ! 

5 

I.O 

!-5 

2.0 

2-5 

3-5 

4.0 

4-5 

60 

6.0 

12.0 

18.0 

24.0 

30.0 

36.0 

42.0 

48.0 

54-0 

6 

0.6 

1.2 

1.8 

2.4 

3-° 

3-6 

4.2 

4.8 

54 

61 

6.1 

12.2 

18.3 

24.4 

30-5 

36.6 

42.7 

48.8 

54-9; 

7 

0.7 

1.4 

2.1 

2.8 

3-5 

4.2 

4.9 

5.6 

6-3 

62 

6.2 

12.4 

18.6 

24.8 

31.0 

37-2 

434 

49.6 

55-8  j 

0.8 

1.6 

2.4 

3-2 

4.0 

4.8 

5.6 

6.4 

7-2 

63 

6-3 

12.6 

18.9 

25.2 

37-8 

44.1 

5°4 

56.7 

9 

0.9 

1.8 

2.7 

3.6 

4-5 

54 

6.3 

7.2 

8.1 

64 

6.4 

12.8 

19.2 

25.6 

32.0 

384 

44.8 

51.2 

57.6 

1O 

I.O 

2.O 

3-° 

4.0 

6.0 

7.0 

8.0 

9.0 

65 

6-5 

13.0 

19-5 

26.0 

32.5 

39-0 

45-5 

52.0 

58.5 

11 

i.i 

2.2 

3-3 

44 

5-5 

6.6 

7-7 

8.8 

9-9 

66 

6.6 

13.2 

19-8 

26.4 

33-° 

39.6 

46.2 

52.8 

594 

i.2 

2.4 

3.6 

4.8 

6.0 

7.2 

8.4 

9.6 

10.8 

67 

6.7 

13-4 

20.1 

26.8 

33-5 

40.2 

46.9 

53-6 

60.3 

13 

J  .-5 

2.6 

3-9 

5-2 

6.5 

7.8 

9.1 

10.4 

11.7 

68 

6.8 

13.6 

20.4 

27.2 

34-o 

40.8 

47.6 

544 

61.2 

14 

1.4 

2.8 

4.2 

5-6 

7.0 

8.4 

9.8 

1  1.2 

12.6 

69 

6.9 

13.8 

20-7 

27.6 

34-5 

41.4 

48.3 

55-2 

62.1 

15 

i-5 

3-° 

4-5 

6.0 

7-5 

9.0 

10.5 

12.0 

13-5 

70 

7.0 

14.0 

21.0 

28.0 

35-o 

42.0 

49.0 

56.0 

63.0 

16 

1.6 

3-2 

4.8 

6.4 

8.0 

9.6 

11.2 

12.8 

14.4 

71 

7-1 

14.2 

21-3 

28.4 

35-5 

42.6 

49-7 

56.8 

63-9 

17 

i-7 

34 

5-1 

6.8 

8-5 

IO.2 

II.9 

I3-6j  15-3 

72 

7-2 

14.4 

21.6 

28.8 

36.0 

43-2 

5°4 

57.6 

64.8 

18 

1.8 

3-6 

54 

7.2 

9.0 

10.8 

12.6 

14.4  16.2 

73 

7-3 

14.4 

21.9 

29.2 

36.5 

43-8 

51-1 

584 

65-7! 

19 

1.9 

3-8 

5-7 

7-6 

9-5 

11.4 

13.3 

15.2  17.1 

74 

.74 

14.8 

22.2 

29.6 

37-o 

444 

51.8 

59-2 

66.6 

20 

2.0 

4.0 

6.0 

8.0 

IO.O 

12.0 

14.0 

1  6.0  1  8.0 

75 

7-5 

15.0 

22-5 

30.0 

37-5 

45>  o 

52-5 

60.0 

67-5 

21 

2.1 

4.2 

6-3 

8.4 

10.5 

12.6 

14.7 

16.8!  18.9 

76 

7.6 

15.2 

22.8 

304 

38.0 

45-6 

53-2 

60.8 

68.4 

22 

2.2 

44 

6.6 

8.8 

II.O 

13.2 

154 

17.61  19.8 

77 

7-7  154 

23.1 

30.8 

38.5 

46.2 

53-9 

61.6 

69-3 

|23 

2-3 

4.6 

6.9 

9.2 

"•5 

I3.8 

16.1 

18.4  20.7 

78 

7.8  15-6 

234 

31.2 

39-o 

46.8 

54.6 

62.4 

70.2 

24 

2.4 

4.8 

7.2 

9.6 

I2.O 

144 

16.8 

19.2121.6 

79 

7-9  15-8 

23.7 

3L6 

39-5 

474 

55-3 

63.2 

71.1 

25 

2-5 

5-° 

7-5 

IO.O 

12.5 

17-5 

2O.O 

22.5 

80 

8.0  16.0 

24.0 

32.0 

40.0 

48.0 

56.0 

64.0 

72.0 

26 

2.6 

5-2 

7-8 

10.4 

13.0 

15.6  18.2 

20.8 

234 

81 

8.1 

16.2 

24-3 

324 

40.5 

48.6 

56.7 

64.8 

72.9 

27 

2.7 

54 

8.1 

10.8 

16.2  18.9 

21.6 

24.3 

82 

8.2 

16.4 

24.6 

32-8 

41.0 

49.2 

574 

65.6 

73-8 

28 

2.8 

5.6 

8.4 

II.2 

I4.0 

i6.8j  19.6 

22.4 

25.2 

83 

8-3 

16.6 

24.9 

33-2 

4i.5 

49.8 

58.1 

66.4 

74-7 

29 

2.9 

5-8 

8.7 

n.6 

14-5 

17.4)20.3 

23.2 

26.1 

84 

8.4 

16.8 

25-2 

33-6 

42.0 

504 

58.8 

67.2 

75-6 

3O 

3-0 

6.0 

9.0 

I2.O 

15.0 

iS.Oj  21.0 

24.0 

27.0 

85 

8.5 

17.0 

25-5 

34-0 

42-5 

51.0 

59-5 

68.0 

76.5 

i 
31 

3-1 

6.2 

9-3 

12.4 

18.6  21.7 

24.8 

27.9 

86 

8.6 

17.2 

25.8 

344 

43-° 

51.6 

60.2 

68.8 

774 

32 

3-2 

6.4 

9.6 

12.8 

16.0 

19.2  22.4 

25.6 

28.8 

87 

8.7 

17.4 

26.1 

34-8 

43-5 

52.2 

60.9 

69.6 

78.3 

33 

3-3 

6.6 

9.9 

13.2 

16.5 

19.8  23.1 

26.4 

29.7 

88 

8.8 

17.6 

26.4 

35-2 

44.0 

52.8 

61.6 

70.4 

79.2 

34 

34 

6.8 

10.2 

13.6 

17.0 

20.4  23.8 

27.2 

30.6 

89 

8-9 

17.8 

26.7 

35-6 

44-5 

534 

62.3 

71.2 

80.  i 

35 

3-5 

7.0 

10.5 

14.0 

17-5 

2  1.  d  24.5 

28.0 

3L5 

90 

9.0 

18.0 

27.0 

36-0 

45-° 

54«o 

63.0 

72.0 

81.0 

36 

3-6 

7-2 

10.8 

14.4 

18.0 

21.6!  25.2 

28.8 

324 

91 

9.1 

18.2 

27.3 

364 

45-5 

54-6 

63-7 

72.8 

81.9 

37 

3-7 

74 

n.  i 

14.8 

18.5 

22.2!  25.9 

29.6 

33-3 

92 

9.2 

18.4 

27.6 

36.8 

46.0 

55-2 

64.4 

73-6 

82.8  ; 

|38 

3-8 

7-6 

11.4 

15.2 

19.0 

22.8  26.6 

304 

34-2 

93 

9-3 

18.6!  27.9 

37-2 

46.5 

55-8 

65.1 

744 

837 

39 
40 

3-9 
4.0 

7-8 
8.0 

11.7 

12.0 

15.6 

16.0 

19-5 

20.0 

234 
24.0 

27-3 
28.0 

31.2 
32.0 

35-i 
36.0 

94 
95 

94 
9-5 

18.8 
19.0 

28.2 
28.5 

37-6 
38.0 

47.0 
47-5 

564 

65.8 
66.5 

75-2 
76.0 

84.6 
85.5 

41 

4.1 

8.2 

12-3 

16.4 

20-5 

24.6 

28.7 

32.8 

36.9 

96 

9.6 

19.2 

28.8 

384 

48.0 

57-6 

67.2 

76.8 

86.4 

42 

4-2 

8.4 

12.6 

16.8 

21.0 

25.2 

29.4 

33.6 

37-8 

97 

9-7 

19.4 

29.1 

38.8 

48.5 

58.2 

67.9 

77-6 

87.3 

43 

4-3 

8.6 

12.9 

17.2 

21.5 

25.8 

30.1 

344 

38.7 

98 

9,8 

19.6 

29.4 

39-2 

49.0 

58.8 

68.6 

78.4 

88.2 

44 

44 

8.8 

13.2 

17.6 

22.O 

26.4 

30-8 

35-2 

39-6 

99 

9-9 

19.8 

29.7 

39-6 

49-5 

594 

69-3 

79-2 

89.1 

45 

4-5 

9.0 

13-5 

1  8.0 

22.5 

27.0 

36.0 

40.5 

100 

IO.O 

20.0 

3O.O 

40.0 

50.0 

60.0 

70.0 

80.0 

90.0 

46 

4.6 

9-2 

13-8 

18.4 

23.0 

27.6 

32.2 

36.8 

41.4 

101 

10.  1 

2O.2 

30.3 

40.4 

50.5 

60.6 

70.7 

80.8 

90.9 

!47 

4-7 

94 

I4.I 

18.8 

23-5 

282 

32-9 

37-6 

42-3 

102 

IO.2 

20-4 

30.6 

40.8 

51.0 

61.2 

71.4 

81.6 

91.8: 

!48 

4.8 

9.6 

144 

19.2 

24.0 

28.8 

33-6 

384 

43-2 

103 

10.3 

20.6 

30-9 

41.2 

61.8 

72.1 

82.4 

92.7  1 

49 

4-9 

9.8 

14.7 

19.6 

24.5 

29.4 

34-3 

39-2 

44.1 

104 

10.4 

20.8 

31.2 

41.6 

52.0 

62.4 

72.8 

83.2 

93-6 

50 

IO.O 

15-0 

20.0 

25.0 

3O.O 

40.0 

45-0 

105 

10.5 

21.0 

31-5 

42.0 

52.5 

63.0 

73-5 

84.0 

94-5 

51 

52 
53 

5-2 
5-3 

10.2 

10^6 

15-3 
15.6 

15-9 

20.4 
20.8 
21.2 

25^ 
26.0 
26.5 

30.6 
31.2 
31.8 

35-7 
364 
37-i 

40.8 
41.6 

424 

45-9 
46.8 

47-7 

1O6 
107 
108 

10.6 
10.7 
10.8 

21.2 
21.4 
21.6 

31-8 
32.1 

32-4 

42.4 
42.8 
43-2 

53-o 
53-5 
54-0 

63.6 
64.2 
64.8 

74-2 
74.9 
75.6 

84.8 
85.6 
86.4 

97-2 

54 

54 

10.8 

16.2 

21.6 

27.0 

324 

37-8 

43-2 

48.6 

1O9 

10.9 

21.8 

32.7 

43-6 

54-5 

654 

76.3 

87.2 

98.1 

55 

5-5 

II.O 

16.5  22.0 

27-5 

33-°  38.5 

44.0 

49.5 

110 

II.O 

22.d  33.0 

44.0 

55-0 

66.0 

77.0*  88.0 

99.0, 

TABLE  XXL 


Squai  «'s  of  Nnmbcrs  increasing  by  Tenths  from  O.O  to  10O.9. 

No. 

O.O 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

09 

Diff. 
0.01  | 

0 

o.o 

0.0 

0.0 

O.I 

O.2 

o-3 

0.4 

o-5 

0.6 

0.8 

O.OI 

1 

I.O 

1.2 

1.4 

1-7 

2.0 

2-3 

2.6 

2.9 

3-2 

3.6 

03 

2 

4.0 

4.4 

4.8 

5-3 

5.8 

6-3 

6.8 

7-3 

7-8 

8.4 

°5 

3 

9.0 

9.6 

10.2 

10.9 

1  1.6 

12.3 

13.0 

'3-7 

14.4 

15.2 

07 

4 

16.0 

1  6.8 

I7.6 

18.5 

19.4 

20.3 

21.2 

22.1 

23.0 

24.0 

09   i 

5 

25.0 

26.0 

27.0 

28.1 

29.2 

30.3 

3M 

32.5 

33-6 

34-8 

O.I  I 

i     6 

36.0 

37-2 

38.4 

39-7 

41.0 

42.3 

43-6 

44.9 

46.2 

47-6 

13 

1      V 

49.0 

504 

51.8 

53-3 

54-8 

56.3 

57-8 

59-3 

60.8 

62.4 

15 

8 

64.0 

65.6 

67.2 

68.9 

70.6 

72-3 

74-o 

75-7 

77-4 

79-2 

17 

9 

81.0 

82.8 

84.6 

86.5 

88.4 

90-3 

92.2 

94.1 

96.0 

98.0 

19 

10 

1  00.0 

IO2.O 

104.0 

1  06.  i 

108.2 

110.3 

112.4 

"4-5 

116.6 

118.8 

0.21 

11 
12 

121.0 

144.0 

123.2 
146.4 

125.4 
148.8 

127.7 
!5i-3 

130.0 
153-8 

132.3 
156.3 

134.6 

158.8 

136.9 
161.3 

igt 

141.6 
166.4 

23 

25 

13 

169.0 

I7I.6 

174.2 

176.9 

179.6 

182.3 

185.0 

187.7 

-    190.4 

193.2 

27    • 

14 

196.0 

198.8 

2OI.6 

204.5 

207.4 

210.3 

213.2 

216.1 

219.0 

222.0 

29  : 

15 

225.0 

228.0 

231.0 

234-1 

237-2 

240.3 

243-4 

246.5 

249.6 

252.8 

0.31 

16 

256.0 

259.2 

262.4 

265.7 

269.0 

272.3 

275.6 

278.9 

282.2 

285.6 

33    ' 

17 

289.0 

292.4 

295.8 

299-3 

302.8 

306-3 

309.8 

3I3-3 

316.8 

320.4 

35   ! 

18 
19 

324.0 

361.0 

327.6 
364-8 

33  i  -2 

368.6 

334-9 
372.5 

338.6 
376.4 

342-3 
380.3 

346.0 
384-2 

349-7 
388.1 

353-4 
392.0 

357-2 
396.0 

37   : 

39    ; 

20 

400.0 

404.0 

408.0 

412.1 

416.2 

420.3 

424.4 

428.5 

432.6 

436.8 

0.41 

21 

441.0 

445-2 

449-4 

453-7 

458.o 

462.3 

466.6 

470.9 

475-2 

479.6 

43 

22 

484.0 

488.4 

492.8 

497-3 

501.8 

506.3 

510.8 

5I5-3 

519.8 

524.4 

45    ' 

23 

529.0 

533-6 

538.2 

542.9 

547-6 

552.3 

557-0 

561-7 

566.4 

571-2 

47 

24 

576.0 

580.8 

585-6 

590.5 

595-4 

600.3 

605.2 

610.1 

615.0 

620.0 

49    i 

25 
26 

625.0 
676.0 

630.0 
681.2 

635-0 
686.4 

640.1 
691.7 

645.2 
697.0 

650.3 
702.3 

6554 
707.6 

660.5 
712.9 

665.6 
718.2 

670.8 
723.6 

0.51    ; 
53 

27 

28 
29 

729.0 
784.0 

841.0 

734-4 
789.6 
84t>.8 

739-8 
795-2 
852.6 

745-3 
800.9 
858.5 

750.8 
806.6 
864.4 

756.3. 
812.3 
870.3 

761.8 
818.0 
876.2 

767-3 

Ilil 

772.8 
829.4 
888.0 

778.4 
835-2 
894.0 

55 
57 

59  ; 

30 

900.0 

906.0 

912.0 

918.1 

924.2 

930.3 

936-4 

942.5 

948.6 

954-8 

0.61 

31 

961.0 

967.2 

973-4 

979-7 

986.0 

992.3 

998.6 

1004.9 

ion.  2 

1017.6 

63 

32 

1024.0 

1030.4 

1036.8 

1043-3 

1049.8 

1056.3 

1062.8 

1069.3 

1075-8 

1082.4 

65 

33 

1089.0 

1095.6 

IIO2.2 

1108.9 

1115.6 

1122.3 

1129.0 

XI35-7 

1142.4 

1149.2 

67   i 

34 

1156.0 

1162.8 

1169.6 

1176.5 

1183.4 

1190.3 

1197.2 

1204.1 

I2II.O 

1218.0 

69    : 

35 

1225.0 

1232.0 

1239.0 

1246.1 

1253.2 

1260.3 

1267.4 

1274-5 

I28I.6 

1288.8 

0.71 

36 

1296.0 

1303.2 

I3I0.4 

I3I7.7  |  1325.0 

1332.3 

I339.6 

1346.9 

1354-2 

1361.6 

73   i 

37 

1369.0 

1376.4 

I383-8 

I39I-3     '398.8 

1406.3 

1413.8 

1421.3 

1428.8 

1436.4 

75    : 

38 

1444.0 

1451.6 

1459.2 

1466.9  1  1474.6 

1482.3 

1490.0 

1497-7 

I505-4 

1513-2 

77 

39 

1521.0 

1528.8 

1536.6 

1544-5  1  1552.4 

1560.3 

1568.2 

1576.1 

1584.0 

1592.0 

79   1 

I 

40 

1  600.0 

1  608.0 

1616.0 

1624.1      1632.2 

1640.3  1  1648.4 

1656.5 

I664.6 

1672.8 

0.8  1 

41 

1681.0 

1689.2 

1697.4 

1705-7 

1714.0 

1722.3  i  1730.6 

1738-9 

1747.2 

!755-6 

83 

42 
43 

1764.0 
1849.0 

1772.4 
1857.6 

1780.8 
1866.2 

1789-3 
1874.9 

1797.8 
1883.6 

1806.3 
1892.3 

1814.8 
1901.0 

1823.3 

1831.8 
1918.4 

1840.4 
1927.2 

85 
87 

44 

1936.0 

1944.8 

1953-6 

1962.5 

1971.4 

1980.3 

1989.2 

1998.1 

2OO7.O 

2016.0 

89   ! 

45 

2025.0 

2034.0 

2043.0 

2052.1 

2061.2 

2070.3 

2079.4 

2088.5 

2097.6 

2106.8 

0.91 

46 

2116.0 

2I25J.2 

2134-4 

2H3.7 

2153.0 

2162.3 

2171.6 

2180.9 

2I9O.2 

2199.6 

93 

47 

2209.0 

2218.4 

2227.8 

2237-3 

2246.8 

2256.3 

2265.8 

2275-3 

2284.8 

2294.4 

95 

48 

2304.0 

23I3.6 

2323.2 

2332-9 

2342.6 

2352-3 

2362.0 

237I-7 

2381.4 

2391.2 

97 

49 

2401.0 

2410.8 

2420.6 

2430-5 

2440.4 

2450-3 

2460.2 

2470.1 

2480.0 

2490.0 

99 

50 

2500.0 

2510.0 

2520.0 

2530.1 

2540.2 

2550-3 

2560.4 

2570-5 

2580.6 

2590.8 

I.OI 

. 

I 



TABLE  XXI. 


91 


Squares  of  Numbers  increasing  by  Tenths  from  O.O  to  1OO.9. 

No. 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

Diff. 
0.01 

50 

2500.0 

2510.0 

2520.0 

2530-1 

2540.2 

2550.3 

2560.4 

2570.5 

2580.6 

2590.8 

I.OI 

51 

2601.0 

2611.2 

2621.4 

2631.7 

2642.0 

2652.3 

2662.6 

2672.9 

2683.2 

2693.6 

°3 

2704.0 

2714.4 

2724.8 

2735-3 

2745-8 

2756.3 

2766.8 

2777-3 

2787.8 

2798.4 

05 

53 

2809.0 

2819.6 

2830.2 

2840.9 

2851.6 

2862.3 

2873.0 

2883.7 

2894.4 

2905.2 

07 

54 

2916.0 

2926.8 

2937.6 

2948.5 

2959.4 

2970-3 

2981.2 

2992.1 

3003.0 

3014.0 

09 

55 

3025.0 

3036.0 

3047.0 

3058.1 

3069.2 

3080.3 

3091.4 

3002.5 

3013.6 

3124.8 

i.  ii 

56 

3136.0 

3H7-2 

3158.4 

3169.7 

3181.0 

3192-3 

3203-6 

3214.9 

3226.2 

3237.6 

13 

57 

3249.0 

3260.4 

3271.8 

3283-3 

3294.8 

3306.3 

3317.8 

3329-3 

3340-8 

33524 

'5 

1     58 

3364-0 

3375-6 

3387-2 

3398.9 

3410.6 

3422.3 

3434-0 

3445-7 

3457-4 

3469.2 

17 

59 

3481.0 

3492-8 

3504.6 

3528.4 

3540-3 

3552-2 

3564-1 

3576.0 

3588.0 

6O 

3600.0 

3612.0 

3624.0 

3636-1 

3648.2 

3660.3 

3672-4 

3684.5 

3696.6 

3708.8 

I.2I 

61 

3721.0 

3733-2 

3745-4 

3757-7 

3770.0 

3782.3 

3794-6 

3806.9 

3819-2 

3831-6 

23 

62 

3844.0 

3856-4 

3868.8 

3881.3 

3893.8 

3906.3 

3918.8 

3931-3 

3943-8 

3956.4 

25 

63 

3969.0 

3981.6 

3994-2 

4006.9 

4019.6 

4032-3 

4045.0 

4057-7 

4070.4 

4083.2 

27 

64 

4096.0 

4108.8 

4121.6 

4134.5 

4147.4 

4160.3 

4I73-2 

4186.1 

4199.0 

4212.0 

29 

65 

4225.0 

4238.0 

4251.0 

4264.1 

4277-2 

4290.3 

4303-4 

43*6-5 

4329.6 

4342.8 

!.3I 

66 

4356.0 

4369.2 

4382.4 

4395-7 

4409.0 

4422.3 

4435-6 

4448.9 

4462.2 

4475-6 

33 

67 

4489.0 

4502.4 

4515-8 

4529-3 

4542.8 

4556.3 

4569.8 

4583-3 

4596.8 

4610.4 

35 

68 

4624.0 

4637-6 

4651.2 

4664.9 

4678.6 

4692.3 

4706.0 

47I9.7 

4733-4 

4747-2 

37 

69 

4761.0 

4774.8 

4788.6 

4802.5 

4816.4 

4830.3 

4844.2 

4858.1 

4872.0 

4886.0 

39 

70 

4900.0 

4914.0 

4928.0 

4942.1 

4956.2 

4970.3 

4984.4 

4998.5 

5012.6 

5026.8 

1.41 

71 

5041.0 

5055-2 

5069.4 

5083.7 

5098.0 

5"2.3 

5126.6 

5140.9 

5I55-2 

5169-6 

43 

72 

5184.0 

5198.4 

5212.8 

5227-3 

5241.8 

5270.8 

5285-3 

5299.8 

45 

73 

5329-0 

5343-6 

5358.2 

5372-9 

5387.6 

5402.3 

54I7-0 

5431-7 

5446.4 

5461.2 

47 

74 

5476.0 

5490.8 

5505-6 

5520.5 

5535-4 

5550.3 

5565-2 

5580.1 

5595-0 

5610.0 

49 

75 

5625.0 

5640.0 

5655-0 

5670.1 

5685-2 

5700.3 

57154 

5730.5 

5745-6 

5760.8 

1.51 

76 

5776.0 

5791-2 

5806.4 

5821.7 

5837-0 

5852-3 

5867.6 

5882.9 

5898.2 

59I3-6 

53 

77 

5929.0 

5944-4 

5959-8 

5975-3 

5990.8 

6006.3 

6021.8 

6037-3 

6052.8 

6068.4 

55 

78 

6084.0  '   6099.6 

6115.2 

6130-9 

6146.6 

6162.3 

6178.0 

6193-7 

6209.4 

6225.2 

57 

79 

6241.0]   6256.8 

6272.6 

6288.5 

6304.4 

6320.3 

6336-2 

6352-1 

6368.0 

6384.0 

59 

80 

6400.0 

6416.0 

6432.0 

6448.1 

6464.2 

6480.3 

6496.4 

6512.5 

6528.6 

6544-8 

1.61 

81 

6561.0 

6577.2 

6593-4 

6609.7 

6626.0 

6642.3 

6658.6 

6674.9 

6691.2 

6707.6 

63 

82 

6724.0 

6740.4 

6756.8 

6773-3 

6789.8 

6806.3 

6822.8 

6839-3 

6*855.8 

6872.4 

.65 

83 

6889.0 

6905.6 

6922.2 

6938-9 

6955-6 

6972.3 

6989.0 

7005.7 

7022.4 

7039.2 

67  ; 

84 

7056.0 

7072.8 

7089.6 

7106.5 

7123.4 

7H0.3 

7I57.2 

7174.1 

7191.0 

7208.0 

69 

85 

7225.0 

7242.0 

7259.0 

7276.1 

7293.2 

73  i  o-3 

73274 

7344-5 

7361.6 

7378.8 

1.71 

86 

7390.0 

74I3-2 

7430.4 

7447-7 

7465.0 

7482.3 

7499.6 

75  i  6.9 

7534-2 

7551-6 

73 

87 

7569.0 

7586.4 

7603.8 

7621.3 

7638.8 

7656-3 

7673.8 

7691-3 

7708.8 

7726.4 

75 

88 

7  744.0  |    7761.6 

7779-2 

7796.9 

7814.6 

7832.3 

7850.0 

7807.7 

7885.4 

7903.2 

V 

89 

7921.0  I    7938.8 

7956.6 

7974-5 

7992.4 

8010.3 

8028.2 

8046.1 

8064.0 

8082.0 

79 

90 

8100.0 

8118.0 

8136.0 

8154-1 

8172.2 

8190.3     8208.4 

8226.5     8244.6 

8262.8 

1.81 

91 

92 
93 

8281.0 
8464.0 
8649.0 

8299.2 
8482.4 
8667.6 

8317-4 
8500.8 
8686.2 

8335-7 
8519-3 
8704.9 

8354.0 
8537-8 
8723.6 

8372-3     8390.6 
8556-3     8574.8 
8742.3     8761.0 

8408.9     8427.2 
8593-3!    8611.8 
8779.7     8798.4 

8445-6 
8630.4 
8817.2 

II 

87 

94 

8836.0 

8854.8 

8873.6 

8892.5 

8911.4 

8930-3 

8949.2 

896  S.i 

8987.0 

9006.0 

89 

95 

9025.0 

9044.0 

9063.0 

9082.1 

9ior.2 

9120.3 

91394 

9158.5 

9177.6 

9196.8 

1.91 

96 

9216.0 

9235-2 

9254.4)   9273.7 

9293.0 

93  i  2.3     933  i  -6 

9350-9  i    9370.2 

9389-6 

93 

97 

9409.0 

9428.4 

9447.8!    9467.3 

9486.8 

9506.3     9525.8 

9545-3     9564-8 

9584.4 

95 

98 

9604.0 

9623.6 

9643.2  !  9662.9 

9682.6 

9702.3!   9722.0     9741.7 

9701.4 

9781.2 

97 

99 

9801.0 

9820.8 

9840.6  1  9860.5 

9880.4 

9900.3  i   9920.2     9940.1     9960.0 

9980.0 

1.99 

i 

r 

100 

IOOOO.O 

IOO2O.O 

10040.0  j  10060.  i 

10080.2 

10100.3 

10120.4 

10140.5 

10160.6 

10180.8 

2.01 

92 


TABLE  XXII. 


Square  Roots  of  Numbers  increasing  by  Tenths  from  O.O  to  1OO.9. 

No. 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

Diff. 
0.01 

0 

0.000 

0.316 

0.447 

0.548 

0.632 

0.707 

0.775 

0.837 

0.894 

0.949 

1 
2 

1.  000 

414 

4.049 
449 

1.095 
483 

1.140 

549 

1.225 

581 

1.265 
612 

1.304 
643 

1.342 
673 

I.378. 
703 

4-15 

3 

732 

761 

789 

817 

871 

897 

924 

95° 

975 

2.69 

4 

2.000 

2.025 

2.049 

2.074 

2.098 

2.I2I 

2.145 

2.168 

2.191 

2.214 

2-37 

5 

2.236 

2.258 

2.280 

2.302 

2.324 

2.345 

2.366 

2.387 

2.408 

2.429 

2.14 

6 

449 

470 

490 

510 

53° 

550 

569 

588 

608 

627 

1.97 

7 

9 

646 
828 
3.000 

665 
846 
3.017 

683 
864 
3-033 

702 
88  1 
3-050 

720 
898 
3.066 

739 
915 
3.082 

757 
3.098 

775 
95° 
3-iH 

966 
3-130 

811 

983 
3.146 

1.83 
1.72 
1.63 

10 

3.162 

3.178 

3-J94 

3.210 

3-225 

3-240 

3-256 

3.271 

3.286 

3-302 

i-55 

11 

317 

332 

347 

362 

376 

391 

406 

421 

435 

45° 

48 

13 

464 
606 

479 
619 

493 
633 

647 

521 
661 

536 
674 

688 

564 
701 

578 
715 

592 
728 

42 
36 

14 

742 

755 

768 

782 

795 

808 

821 

834 

847 

860 

32 

15 

3-873 

3.886 

3-899 

3.912 

3-924 

•3-937 

3-950 

3-962 

3-975 

3.987 

1.27 

16 

4.000 

4.012 

4-025 

4-037 

4.050 

4.062 

4.074 

4.087 

4.099 

4.111 

23 

17 

123 

135 

159 

171 

183 

195 

207 

219 

231 

20 

18 

243 

254 

266 

278 

290 

301 

3*3 

324 

336 

348 

16 

19 

359 

370 

382 

393 

405 

416 

427 

438 

450 

461 

13 

20 

4-472 

4-483 

4-494 

4.506 

4-5I7 

4-528 

4-539 

4-550 

4.561 

4-572 

1.  10 

21 

583 

593 

604 

615 

626 

637 

658 

669 

680 

08 

22 

690 

701 

712 

.    722 

733 

743 

754 

764 

775 

785 

06 

23 

796 

806 

817 

827 

837 

848 

858 

868 

879 

889 

°3 

24 

899 

909 

919 

93° 

940 

950 

960 

970 

980 

990 

OI 

25 

5.000 

S-oio 

5.020 

5-030 

5.040 

5-050 

5.060 

5-070 

5-079 

5-089 

0.99 

26 

099 

109 

119 

128 

138 

148 

I58 

167 

177 

187 

97 

27 

28 

196 
292 

206 
301 

215 
310 

225 

320 

235 
329 

244 
339 

348 

263 
357 

273 
367 

282 
376 

95 
94 

29 

385 

394 

4°4 

422 

441 

450 

459 

468 

92 

30 
31 

51S 

5.486 

•577 

5-495 

5-505 

S-SH 
604 

5-523 
612 

5-532 
621 

630 

5-550 
639 

5-559 
648 

0.91 

32 

657 

666 

675 

683 

692 

701 

710 

718 

727 

736 

88 

33 

745 

753 

762 

771 

779 

788 

797 

805 

814 

822 

87 

34 

831 

840 

848 

857 

865 

874 

882 

891 

899 

908 

85 

35 
36 

5.916 
6.000 

5-925 
6.008 

5-933 
6.017 

5-941 
6.025 

5-95° 
6.033 

5.958 
6.042 

5-967 
6.050 

5-975 
6.058 

Ifel 

5-992 
6.075 

0.85 
83 

37 

083 

091 

099 

107 

116 

124 

132 

140 

'148 

156 

82 

38 

164 

173 

181 

189 

197 

205 

213 

221 

229 

237 

81 

39 

245 

253 

261 

269 

277 

285 

293 

3OI 

309 

80 

40 

6-325 

6-332 

6.340 

6-348 

6.356 

6.364 

6-372 

6.380 

6.387 

6-395 

0.79 

41 

403 

411 

419 

427 

434 

442 

450 

458 

465 

473 

78 

42 
43 
44 

481 
557 
633 

488 

496 

III 

504 
580 
656 

512 

588 
663 

519 
595 
671 

67^ 

535 
611 
686 

542 

.    618 
693 

550 
626 
701 

77 
76  1 
75  | 

45 
46 

6.708 
782 

6.716 

790 

6.723 

797 

6.731 
804 

6.738 
812 

6-745 
819 

6-753 
826 

6.760 
834 

6.768 
841 

6ill 

0.74 
73 

47 

856 

863 

870 

877 

885 

892 

899 

907 

914 

921 

73 

48 

928 

935 

943 

95° 

957 

964 

971 

979 

986 

993 

72 

49 

7.000 

7.007 

7.014 

7.021 

7.029 

7.036 

7-043 

7.050 

7-057 

7.064 

7i 

50 

7.071 

7-078 

7.085 

7.092 

7.099 

7.106 

7-"3 

7.120 

7.127 

7-134 

0.70 

TABLE  XXII. 


Square  Roots  of  Numbers  increasing:  by  Tentlts  front  O.O  to  1OO.9. 

No. 

0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

O.9 

Diff. 
0.01 

. 

50 

7.071 

7.078 

7.085 

7.092 

7.099 

7.106 

7-113 

7.120 

7.127 

7.134 

0.70 

51 

141 

148 

155 

162 

169 

176 

183 

190 

197 

204 

70 

52 

211 

218 

225 

232 

239 

246 

253 

259 

266 

273 

69 

53 

280 

287 

294 

301 

308 

314 

321 

328 

335 

342 

68 

54 

348 

355 

362 

369 

376 

382 

389 

396 

403 

409 

68 

55 

7.416 

7-423 

7-43° 

7-436 

7-443 

7-45° 

7-457 

7463 

7.470 

7.477 

0.67 

56 

483 

490 

497 

5°3 

510 

517 

523 

530 

537 

543 

67 

57 

55° 

556 

563 

570 

576 

583 

589 

596 

603 

609 

66  ! 

58 

616 

622 

629 

635 

642 

649 

655 

662 

668 

675 

65 

59 

681 

688 

694 

701 

707 

7H 

720 

727 

733 

74° 

65i 

6O 

7.746 

7-752 

7-759 

7-765 

7.772 

7.778 

7.785 

7.791 

7-797 

7.804 

0.64 

61 

810 

817 

823 

829 

836 

,  842 

849 

855 

86  1 

868 

64i 

62 

874 

880 

887 

893 

899 

906 

912 

918 

925 

93  i 

63 

63 
64 

937 

8.000 

944* 
8.006 

950 

8.012 

956 
8.019 

962 
8.025 

969 
8.031 

975 
8-037 

981 
8.044 

987 
8.050 

994 
8.056 

63 

62  i 

65 

8.062 

8.068 

8.075 

8.081 

8.087 

8.093 

8.099 

8.106 

8.II2 

8.118 

0.62  ; 

66 

124 

130 

136 

142 

149 

'55 

161 

167 

173 

179 

61  i 

67 

185 

191 

198 

204 

210 

216 

222 

228 

234 

240 

61  1 

68 

246 

252 

258 

264 

270 

276 

283 

289 

295 

301 

60 

69 

307 

313 

319 

325 

331 

337 

343 

349 

355 

60 

7D 

8.367 

8-373 

8-379 

8.385 

8.390 

8.396 

8.402 

8.408 

8.414 

8.420 

O.6O   ; 

71 

426 

432 

438 

444 

45° 

456 

462 

468 

473 

479 

59  > 

72 

485 

491 

497 

5°3 

509 

515 

.521 

526 

532 

538 

59  ; 

73 

544 

•  55° 

556 

562 

567 

573 

579 

585 

597 

58- 

74 

602 

608 

614 

620 

626 

631 

637 

643. 

649 

654 

58  ; 

75 

8.660 

8.666 

-8.672 

8.678 

8.683 

8.689 

8.695 

8.701 

8.706 

8.712 

0.58 

76 

718 

724 

729 

735 

74* 

746 

752 

758 

764 

769 

57 

77 

775 

781 

786 

792 

798 

803 

809 

815 

820 

826 

57 

78 

o  _ 

837 

843 

849 

854 

860 

866 

871 

877 

883 

56 

79 

888 

894 

899 

905 

911 

916 

922 

927 

933 

939 

56  i 

89 

8.944 

8.950 

8-955 

8.961 

8.967 

8.972 

8.978 

8.983 

8.989 

8.994 

0.56 

81 

9.000 

9.006 

9.011 

9.017 

9.022 

9.028 

9-033 

9-039 

9-044 

9.050 

56; 

82 

055 

061 

066 

072 

077 

083 

088 

094 

099 

I05 

55 

83 

no 

116 

121 

127 

132 

138 

143 

149 

154 

1  60 

55 

84 

165 

171 

I76 

182 

187 

192 

198 

203 

209 

214 

54 

85 

9.220 

9-225 

9.230 

9.236 

9.241 

9.247 

9.252 

9-257 

9.263 

9.268 

°-54  : 

88 

274 

279 

284 

290 

295 

301 

306 

311 

322 

54 

87 

327 

338 

343 

349 

354 

359 

365 

37o 

375 

53 

88 

381 

386 

391 

397 

402 

407 

418 

423 

429 

53  : 

89 

434 

439 

445 

45° 

455 

460 

466 

471 

476 

482 

53 

90 

9.487 

9-492 

9-497 

9-5°3 

9.508 

9.513 

9.518 

9-524 

9-529 

9-534 

o-53 

91 

539 

545 

55° 

555 

560 

566 

576 

581 

586 

52  : 

92 

592 

597 

602 

607 

612 

618 

623 

628 

633 

638 

52 

93 

644 

649 

654 

659 

664 

670 

675 

680 

685 

690 

52  1 

94 

695 

701 

706 

711 

716 

721 

726 

73  i 

737 

742 

51 

95 

9-747 

9-752 

9-757 

9.762 

9.767 

9.772 

9.778 

9.783 

9-788 

9-793 

0.51 

96 

798 

803 

808 

813 

818 

823 

829 

834 

839 

844 

51 

97 

849 

854 

859 

864 

869 

874 

879 

884 

889 

894 

51 

98 

899 

905 

910 

920 

925 

930 

935 

940 

945 

99 

950 

955 

960 

965 

970 

975 

980 

985 

990 

995 

50  ; 

100 

IO.OOO 

10.005 

IO.OIO 

10.015 

IO.O2O 

10.025 

10.030 

10.035 

10.040 

10.045 

0.50 

94 


TABLE  XXIII. 


True  Rising*  and  Setting*. 

Declination  of  same  name  as  the  Latitude. 

<D 

1 

0° 

1° 

2 

3° 

4° 

5° 

6° 

•§ 

J 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris.    Set. 

Ris. 

Set. 

«    | 

" 

1 

"    i 

0 

ft  m 

ft  m 

ft     TO 

ft    in 

ft    TO 

ft  m 

ft    TO 

ft    TO 

ft   m  I  ft  TO 

ft  m    It   m 

ft     TO 

ft     TO 

O     ' 

0 

6  o 

6  o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6060 

6060 

6    o 

6    o 

0 

4 

o 

0 

0 

o 

5  59 

i 

5  59 

i 

5  59  I       i 

559         i 

5  58 

2 

4 

8 

0 

o 

5  59 

i 

CQ 

i 

58 

2 

58 

2 

57!       3 

57 

3 

8 

'•  12 

o 

0 

59 

i 

58 

2 

57 

3 

57 

3 

56  :       4 

55 

5 

12 

16 

o 

o 

59 

i 

58 

2 

57 

3 

55  1       5 

54 

6 

53 

7 

16 

i  20 

6  o 

6  o 

5  59 

6     i 

5  57 

6    3 

5  56 

6    4 

5  54  6    6 

5  53 

6     7 

5  51 

6    9 

2O 

22 

o 

o 

58 

2 

57 

3 

55 

5 

54  1       6 

52         8 

50 

10 

22 

24 

0 

o 

58 

2 

56 

4 

55 

5 

53         7 

51  i    9 

49 

ii 

24 

26 

o 

0 

58 

2 

56 

4 

54        6 

52         8 

50       10 

48 

12 

26 

28 

o 

0 

58 

2 

56 

4 

54         6 

51         9 

49 

ii 

47 

13 

28 

3f> 

6  o 

6  o 

5  58 

6      2 

5  55 

6    5 

5  53   6    7 

5  51 

6    9 

5  48 

6  12 

5  46 

6  14 

3O 

32 

o 

0 

58 

2 

55 

5 

53         7 

50 

IO 

47 

13 

45 

15 

32 

34 

0 

o 

57 

3 

55 

52        8 

49 

ii 

46       14 

44 

16 

34 

:  36 

o 

o 

57 

3 

54 

6 

51 

9 

48 

12 

45 

15 

42 

18 

36 

!  38 

o 

0 

57 

3 

54 

6 

9 

47 

13 

44 

16 

19 

38 

40 

6  o 

6  o 

5  57 

6    3 

5  53 

6     7 

5  50 

6  10 

5  47 

6  13 

5  43   6  17 

5  40 

6   20 

4O 

41 

o 

0 

57 

3 

53 

7 

5° 

10 

46 

14 

43 

17 

39 

21 

41 

42 

0 

o 

56 

4 

53 

7 

49 

ii 

46 

14 

42 

18 

38 

22 

42 

!  43 

o 

o 

56 

4 

53 

7 

49 

ii 

45 

15 

41 

19 

38 

22 

43 

44 

o 

o 

56 

4 

52 

8 

48 

12 

45 

15 

41        19 

37 

23 

44 

45 

6  o 

6  o 

5  56 

6    4 

5  S2 

6    8 

548 

6    12 

5  44 

6  16 

5  40 

6   20 

5  36 

6  24 

45 

46 

0 

o 

56 

4 

52 

8 

48 

12 

43 

17 

21 

35 

25 

46 

47 

0 

o 

56 

4 

51 

9 

47 

13 

43 

17 

38 

22 

34 

26 

47 

48 

o 

•    0 

56 

4 

51 

9 

47 

42 

18 

38 

22 

33 

27 

48 

49 

0 

o 

55 

5 

9 

46 

4 

42 

18 

37 

23 

32 

28 

49 

50 

6  o 

6  o 

5  55 

6    5 

5  5° 

6  10 

5  46 

6  14 

5  41 

6  19 

536 

6  24 

5  31 

629 

5O 

51 

0 

o 

55 

5 

5° 

IO 

45 

15 

40 

20 

35 

25 

30 

3° 

51 

52 

o 

0 

55 

5 

.  10 

45 

15 

39 

21 

34 

26 

29 

31 

52 

53 

0 

o 

55 

5 

49 

ii 

44 

16 

39 

21 

33 

27 

28 

32 

53  i 

54 

o 

o 

55 

5 

49 

ii 

44 

16 

38 

22 

32 

28 

27 

33 

51 

i  55 

6  o 

6  o 

5  54 

6    6 

5  49 

6  ii 

5  43 

6  17 

5  37 

623 

5  31 

6  29 

5  25 

635 

55 

56 

o 

0 

54 

6 

48 

12 

42 

18 

36 

24 

3<> 

30 

24 

36 

56 

i  57 

o 

o 

54 

6 

48 

12 

41 

19 

35 

25 

29      3i 

23 

37 

57  ! 

:  58 

0 

o 

54 

6 

47 

IT 

41 

19 

34 

26 

28 

32 

21 

39 

58 

59 

o 

o 

53 

7 

47 

13 

40 

20 

33 

27 

27 

33 

20 

40 

59 

60 

6  o 

6  o 

5  53 

6    7 

5  46 

6  14 

5  39 

6   21 

5  32 

6  28 

5  25 

635 

5  18 

6  42 

60 

61 

o 

o 

53 

46 

14 

38 

22 

29 

24 

36 

16 

44 

61 

62 

o 

0 

52 

8 

45 

15 

37 

23 

30 

3° 

22 

38 

14 

46 

62 

63 

o 

o 

52 

8 

44 

16 

36 

24 

28 

32 

2O 

40 

12 

48 

63 

64 

o 

o 

52 

8 

44 

16 

35 

25 

27 

33 

19 

41 

IO 

50 

64 

65 

6  o 

6  o 

5  51 

6    9 

5  43 

6  17 

5  34 

6  26 

5  26 

634 

5  18 

6  42 

5    8 

652 

65 

66 

o 

o 

51 

9 

42 

18 

33 

27 

24 

36 

IS 

45 

5 

55 

66 

67 

o 

0 

51 

9 

41 

19 

32 

28 

22 

38 

12 

48 

3 

57 

67 

i  68 

0 

o 

5° 

10 

40 

20 

3° 

30 

20 

40 

IO 

5° 

0 

7    o 

68 

69 

o 

0 

IO 

39 

21 

29 

31 

18 

42 

7 

53 

4  56 

4 

69  ! 

70 

6  o 

6  o 

5  49 

6  ii 

5  38 

6  22 

5  27 

633 

5  16 

6  44 

5    4 

6  56 

4  53 

7    7 

70 

71 

o 

o 

48 

12 

37 

23 

25 

35 

13 

47 

i 

59 

49 

ii 

71 

72 

o 

0 

48 

12 

35 

25 

23 

37 

10 

50 

4  58 

7    2 

45 

15 

72 

73 

o 

o 

47 

13 

34 

26 

21 

39 

7 

53 

53 

7 

40 

20 

73 

74 

o 

o 

46 

14 

32 

28 

18 

42 

4 

56 

49 

ii 

34 

26 

74 

75 

6  o 

6  o 

5  45 

6  15 

5  30 

6  30 

5  15 

645 

c     o 

7    o 

4  44 

7  16 

4  28 

7  32 

75 

76 

0 

o 

44 

16 

28 

32 

ii 

49 

4  55 

5 

38 

22 

20 

40 

76 

77 

o 

o 

43 

17 

25 

35 

7 

53 

49 

ii 

31 

29 

ii 

49 

77 

78 

o 

0 

19 

22 

38 

3 

57 

43 

17 

23 

37 

i 

o  59 

78 

179 

0 

0 

39 

21 

19 

4  57 

7    3 

36 

24 

13 

47 

3  49 

8  u 

79 

r 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set.    Ris. 

6 

0) 

| 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

1 

| 

Declination  of  contrary  name. 

1 

TABLE  XXIII. 


95 


1 
True  Rising-  aiicl  Setting-. 

i 
o> 

Declination  of  same  name  as  the  Latitude. 

. 

•0 

6 

7° 

8° 

9° 

10° 

11° 

12° 

'! 

3 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

nS 

i         O 

hm 

hm 

h   m  \  h   m 

h  m 

h   m 

h   m 

h   m 

h   m 

/*   m 

h   m 

h  m 

h  m 

h    m 

0 

0 

6060 

6    06    o 

6    o 

6    o 

6060 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

0 

4 

5  58  \      2 

5  58  i       2 

5  58 

2 

5  57 

3 

5  57 

3 

S  57 

3 

5  57 

3 

4 

8 

57        3 

56  1        4 

55 

5 

55 

54 

6 

54 

6 

53 

7 

8 

12 

55'      5 

54         6 

53 

7 

52 

8 

9 

9 

10 

12 

16 

53!      7 

52;       8 

9 

50 

10 

48 

12 

47        13 

46 

H 

16  1 

2O 

551  !6  9 

5  5°;  6  10 

S  48 

6    12 

5  47 

6  13 

5  45 

6  15 

S  44 

6  16 

5  42 

6  18 

20  1 

22 

50      10 

49       n 

47 

13 

45 

VS 

44 

16 

42 

18 

40 

20 

22  ! 

24 

49      u 

47       13 

46 

14 

44 

16 

42 

18 

40 

20 

38!         22 

24 

26 

48      12 

46;      14 

44 

16 

42 

18 

40 

20 

38 

22 

36|      24 

26 

28 

47      J3 

45        15 

43 

17 

41 

19 

38 

22 

36 

24 

34        26 

28  ! 

30 

5  46  6  14 

5  44'  6  16 

5  4i 

6  19 

5  39 

6   21 

S  37 

62S 

5  34 

6  26 

5  32  |  6  28 

30  ; 

32 

45      15 

42  1      18 

40  1     20 

37 

23 

35 

25 

32]      28 

29        3i 

32 

34 

44  i     16 

4i        19 

38 

22 

35 

25 

33 

27 

3°        3° 

27|      33 

34  ! 

36 

42      18 

40       20 

37 

23 

34 

26 

31        29 

28        32 

24        30 

36  i 

>  38 

41       19 

38!        22 

35 

25 

32 

28 

2t>           32 

25        35 

22            38 

38 

40 

5  40  i  6  20 

5  36  i  6  24 

5  33 

6  27 

5  29 

631 

5  26 

6  34 

5    22 

6  38 

5  19    6  4i 

40 

i  41 

39  i    21 

35;     25 

32 

28 

28 

32 

25 

35 

21        39 

17        43 

43  i 

42 

22 

35       25 

31       29 

27 

33 

23 

37 

20  i         40 

1  6        4.4 

42  i 

43 

3s 

22 

34      26 

301     30 

26 

54 

22 

38 

18        42 

14  1      4J 

43 

44 

37 

23 

33       27 

29 

31 

25 

35 

21 

39 

17 

43 

13 

47 

44 

j  45 

536 

624 

5  32 

6  28 

^28 

632 

5  24 

6  36 

5  19 

641 

S  IS 

6  45 

5  n 

649 

45 

46 

35 

2S 

3i 

29 

27 

33 

22 

38 

18 

42 

14 

46 

9 

46  ! 

|47 

34 

26 

3°       30 

25 

35 

21 

39 

16 

44 

12 

48 

7 

53 

47  ! 

33 

27 

29 

31 

24 

3§ 

19 

IS 

45 

10 

5° 

5 

55 

48  j 

49 

28 

28 

32 

23 

37 

18 

42 

13 

47 

8 

52 

3!      57 

49 

50 

5  3i 

6  29 

526 

6  34 

5  21 

6  39 

5  i6 

6  44 

5  ii 

6  49 

5    6 

6  54 

5     i 

659 

50  ' 

51 

3° 

3° 

2,S 

35 

20 

40 

IS 

4S 

10 

So 

A 

4  59 

7     i 

51  i 

52 

29 

31 

24 

19 

41 

13 

47 

8 

52 

2 

58 

57 

52  ! 

53 

28 

32 

22 

38 

17 

43 

ii 

49 

6 

54 

oj  7    o 

54         6 

53 

54 

27 

33 

21 

39 

15 

45 

10 

50 

4 

56 

4  58!        2 

52!        8 

54  ! 

55 

525 

63S 

S    20 

6  40 

5  ^4 

6  46 

S    8 

652 

5    2 

6  58 

4  s6 

7    4 

4  49 

7  ii 

55  i 

56 

24 

35 

18 

42 

12 

48 

6 

54 

4  59 

7     i 

53 

7 

47 

13 

56 

57 

23 

37 

16 

44 

10 

50 

4 

57 

So 

10 

44 

16 

57 

58 

21 

39 

IS 

45 

8 

52 

i 

59 

54 

6 

48 

12 

19 

58  j 

59 

20 

40 

13 

47 

6 

54 

4  59 

7     i 

52 

8 

44 

16 

37 

23 

59  ; 

60 

518 

642 

S  ii 

6  49 

5    4 

6  56 

4  56 

7    4 

4  49 

7  u 

4  41 

7  19 

4  34 

7  26 

6O 

61 

16 

44 

9 

i       59 

54 

6 

4'o 

H 

38 

22 

3° 

30 

61 

62 

14 

4b 

7 

53 

4  59 

7    i 

9 

43 

17 

34 

26 

26 

34 

62 

63 

12 

48 

4 

56 

S6 

4 

48 

12 

39 

21 

3° 

3° 

21 

39 

63 

64 

10 

50 

2 

58 

53 

7 

44 

16 

35 

25 

26 

34 

17 

43 

64  j 

65 

5   8 

652 

C        O 

7    o 

4  5° 

7  10 

4  41 

7  19 

4  31 

7  29 

4  21 

7  39 

4  12 

7  48 

65 

66 

5 

55 

4$6 

4 

46 

14 

37 

23 

27 

33 

16 

44 

6 

54 

66 

6? 

3 

57 

53 

7 

43 

17 

33 

27 

22 

38 

ii 

49 

o 

8    o 

67 

68 

o 

7  o 

49 

u 

39 

21 

28 

32 

16 

44 

5 

55 

3  53 

7 

68  ! 

69 

45<> 

4 

45 

15 

34 

26 

23 

37 

II 

49 

3  58 

8      2 

46 

H 

69 

70 

453 

7   7 

4  4i 

7  19 

4  29 

7  31 

4  17 

743 

4    4 

756 

3  5 

8    9 

3  37 

823 

70 

71 

49 

ii 

3& 

24 

23 

37 

10 

50 

3  57 

8    3 

43 

17 

28 

32 

71 

72 

45 

15 

31 

29 

17 

43 

3 

57 

49 

ii 

33!      27 

17 

43 

72 

73 

40 

20 

25 

35 

10 

So 

3  SS 

8    S 

39 

21 

22 

38 

4 

so 

73 

74 

34 

26 

19 

3 

57 

46 

H 

28 

32 

9 

51 

2  49 

9  ii 

74 

75 

428 

732 

4  u 

7  49 

3  54 

8    6 

3  35 

8  2<; 

3  15 

8  45 

2  54 

9    6 

2   30 

9  3° 

75 

76 

20 

40 

2 

58 

43 

17 

22 

38 

o 

O      O 

35 

25 

6 

54 

76 

77 

II 

49 

3  51 

8    9 

3° 

3° 

71     53 

2  41 

19 

ii        49 

i  32 

10  28 

77 

78 

I 

59 

39 

21 

14 

46 

247 

9  13 

16 

44 

i  35  10  25 

0      0 

12      0 

78 

79 

349 

811 

23 

37 

2  55 

9    5 

22 

38 

i  40 

10   20 

0      OJI2      0 

79 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

$ 

o> 
•O 

9 

6° 

7° 

8° 

9° 

10° 

11° 

12° 

£ 

•M 

Declination  of  contrary  name. 

3 

TABLE  XXIII. 


True  Rising  and  Setting-. 

Declination  of  same  name  as  the  Latitude. 

0) 

B 

"§ 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

'O 

•t« 

j*^ 

I 

Ris. 

Set. 

Ris 

Set 

Ris.    Set. 

Ris. 

Set. 

Ris.    Set. 

Ris.    Set. 

Ris.    Set. 

u 

i 

o 

TwT 

h  ni 

h   in 

h  m 

h    m    k   m 

//    ni 

h  m 

h   m 

h   >n 

h   in    k   m 

h    m  \  h    in 

0 

6   o 

6   o 

6    o 

6    o 

6    oi  6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    06     o 

O 

2 

558 

2 

558 

2 

5  58         2 

5  58 

2 

558 

2 

558 

2 

5  57  f       3 

2 

4 

57 

3 

56 

4 

56         4 

56 

4 

55 

5 

55         5 

55i        5 

4 

6 

55 

5 

54 

6 

54 

6 

54 

6 

53 

7 

53 

7 

52          8 

6 

8 

53 

7 

53 

7 

52 

8 

9 

9 

50 

10 

50         10 

8 

10 

5  51 

6   9 

5  51 

6    9 

5  50;  6  10 

5  49 

6  ii 

548 

6    12 

548 

6    12 

5  47    6  13 

10 

12 

50 

10 

49 

ii 

48  j          12 

47 

IT 

46 

*4 

45 

15 

44        16 

12 

14 

48 

12 

47 

13 

46;          14 

45 

15 

44 

16 

43 

17 

41  1       19 

14 

16 

46 

H 

45 

15 

44        16 

42 

18 

41 

19 

40 

20 

39        21 

1G 

18 

44 

16 

43 

41  j       19 

40 

20 

39 

21 

37 

23 

36        24 

18 

20 

542 

618 

5  41 

6  19 

5  39J  6  21 

538 

6   22 

5  36  |  6  24 

5  34 

6  26 

5  33;  6  27 

20 

24 

40 
38 

20 

22 

39 
36 

21 

24 

37  i       23 
34!       26 

35 
33 

25 
27 

33        27 
31        29 

32 
29 

28 
31 

30  i      30 

27'      33 

22 
21 

26 

36 

24 

34 

26 

32        28 

30 

3° 

28 

32 

26 

54 

24'      36 

26 

28 

34 

26 

32 

28 

30        30 

27 

33 

25 

35 

23 

37 

20 

40 

28 

J  30 

532 

628 

5  29 

631 

5  27    6  33 

5  24    6  36 

5    22 

6  38 

5  19 

6  41 

5  17 

6  43 

30 

:  32 

29 

31 

27 

33 

24        36 

21  j         ,Q 

19 

41 

16 

44 

'3 

47 

32 

,  34 

27 

33 

24 

36 

21  ;      39 

18 

42 

15 

45 

12 

48 

9 

34 

36 

24 

3o 

21 

39 

18 

42 

15 

45 

12 

48 

9 

51 

Si      55 

36 

38 

22 

38 

18 

42 

15 

45 

12 

48 

8 

52 

5 

55 

1        59 

38 

40 

519 

641 

5  15 

645 

5  12 

6  48 

5    8    6  52 

5    4 

6  56 

5     i 

6  59 

4  57    7    3 

40 

41 

142 

II 

43 
44 

H 

12 

4* 
48 

10 

8 

50 

52 

6 
4 

54 
56 

2           58 

07    o 

458 
56 

7    2 
4 

54 

52 

6 

8 

41 
42 

'43 

H 

46 

10 

50 

6 

54 

2 

58 

458      '       2 

54 

6 

49 

ii 

43 

i  44 

13 

47 

8 

52 

4 

56 

070 

56          4 

51 

9 

47        13 

44 

'45 

5" 

649 

5     7 

653 

5     2 

6  58 

4  58  !  7    2 

4  53  |  7    7 

4  49    7  11 

4  44    7  16 

45 

46 

9 

51 

5 

55 

0)70 

56 

4 

51          9 

46 

14 

41 

19 

4G 

47 

7 

53 

3 

57 

4  58t        2 

53 

7 

48  1      12 

43 

17 

38 

22 

47 

,48 

5 

55 

i 

59 

56 

4 

51 

9 

46 

H 

19 

35 

25 

48 

;49 

3 

57 

4  58 

7    2 

53 

7 

48 

12 

43 

17 

38 

22 

32 

28 

49 

50 

5    i 

659 

456 

7    4 

4  51 

7    9 

4  46 

7  *4 

4  40 

7  20 

4  35 

7  25 

4  29 

7  31 

50 

51 

459 

7    * 

54 

6 

48 

12 

43 

17 

37 

23 

29 

25 

35 

51 

52 

57 

51 

9 

46 

14 

40 

20 

34 

26 

28 

32 

22 

38 

52 

53 

54 

6 

49 

ii 

43 

17 

37 

23 

31 

29 

24 

36 

18 

42 

53 

54 

52 

8 

46 

J4 

40 

20 

33 

27 

27 

33 

20 

40 

J4 

46 

54 

55 

449 

711 

4  43 

7  17 

4  37 

7  23 

.4  3° 

7  3° 

4  23 

7  37 

4  16 

7  44 

4    9 

7  51 

55 

56 

47 

40 

20 

33 

27 

26 

34 

19 

12 

48 

5 

55 

56 

57 

44 

16 

37 

23 

3° 

3° 

22 

38 

15 

45 

8 

52 

o 

8    o 

57 

58 

41 

19 

33 

27 

26 

34 

18 

42 

ii 

49 

3 

57 

3  55 

5 

58  i 

59 

37 

23 

3° 

30 

22 

38 

14 

46 

6 

54 

358 

8      2 

49 

ii 

59 

60 

434 

726 

4  26 

7  34 

4  18 

7  42 

4    9 

7  51 

4     i 

7  59 

3  52 

8    8 

3  43    8  17 

60 

61 

3° 

30 

22 

38 

47 

4 

56 

3  55 

8    5 

46 

14 

36       24 

61 

62 

26 

34 

17 

43 

8 

52 

3  59 

8    i 

49 

ii 

40 

20 

291      31 

62 

63 

21 

39 

12 

48 

3 

57 

53 

7 

43 

17 

33 

27 

22 

38 

63 

64 

17 

43 

7 

53 

3  57 

8    3 

47 

13 

36 

24 

25 

35 

13 

47 

64 

65 

4I2 

748 

4     i 

7  59 

3  51 

8    9 

3  40 

8   20 

328 

832 

3  16 

8  44 

3    3 

857 

65 

66 

6 

54 

3  54 

8    6 

44 

16 

32 

28 

20 

40 

7 

53 

2  53 

9    7 

66 

67 

o 

8  o 

48 

12 

36 

24 

23 

37 

10 

5° 

2   56 

9    4 

40 

20 

67 

68 

353 

7 

19 

28 

32 

14 

46 

2  59 

9     i 

43 

17 

26 

34 

6§  ; 

69 

46 

14 

32 

28 

18 

42 

3 

57 

46 

14 

29 

31 

9 

5' 

69 

70 

337 

823 

3  23 

837 

3    7 

853 

2  50 

9  10 

2  32 

9  28 

2    12 

948 

i  47 

10  13 

7O 

71 

28 

32 

12 

48 

2  54 

9    6 

36 

24 

15 

45 

I    5° 

10    10 

43 

71 

72 

17 

43 

2  59 

9     i 

40 

20 

18 

42 

I    52 

10    8 

I9|        41 

O      O    12      O 

72 

73 

4 

56 

44 

16 

21 

39 

i  55 

'o    5 

21 

39 

O     O  12     O 

73 

74 

249 

911 

25 

35 

I    58 

10      2 

23 

37 

0      0 

12      0 

74    ; 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set.    Ris. 

1 

0) 

0) 

? 

12° 

' 

13° 

14° 

15° 

16° 

17° 

18° 

3 

•5 

•"S 

01 

Declination  of  contrary  name. 

Li 

TABLE  XXIII. 


97 


True  Rising:  and  Setting:. 

Declination  of  same  name  as  the  Latitude. 

.    .. 

<L> 

o3 

•o 

T3 

3 

18° 

193 

20° 

21° 

22° 

23° 

24° 

5 

!      ^4 

•H 

\      * 

Ris.  I  Set. 

Ris. 

Set. 

Ris.    Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

nt 

O 

h  m  \  h  m 

h   m 

//    nt 

h  in 

h  m 

h   m 

h    m 

h  m 

h    lit 

h  m 

h   m 

h  m    h  m 

o 

0 

6060 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    0160 

6060 

O 

557J      3 

S  57 

3 

5  57 

3 

5  57 

3 

5  57 

3 

5  57         3 

5  56i        4 

2 

4 

55       5 

54 

6 

54 

6 

54 

6 

54 

6 

53 

7 

53 

7 

4 

6 

S2 

8 

S2 

8 

Si 

9 

Si 

9 

5° 

10 

5° 

10 

49 

ii 

6 

8 

50 

IO 

49 

ii 

48 

12 

48 

12 

47 

13 

46 

14 

46 

H 

8 

IO 

547  613 

5  4^ 

6  14 

S  4S 

6  IS 

5  44 

6  16 

5  44 

6  16 

5  43 

6  17 

5  42 

6  18 

10 

12 

44      16 

43 

17 

42 

18 

19 

40 

20 

39 

21 

38 

22 

12 

14 

41       19 

40 

20 

39 

21 

38 

22 

37 

23 

36 

24 

35 

25 

14 

16 

39      21 

37 

23 

36 

24 

35 

25 

33 

27 

32 

28 

29 

16 

18 

36      24 

34 

26 

33 

27 

29 

3° 

30 

28 

32 

27 

33 

18 

2O 

533   627 

5  3i 

6  29 

5  30 

6  30 

528 

6  32 

5  26 

6  34 

5  24    6  36 

S  23 

637 

20 

22 

3°      30 

28 

32 

26 

34 

24 

36 

22 

38 

20  1      40 

19 

22 

24 

27 

33 

25 

35 

23 

37 

21 

39 

19 

i6|       44 

46 

24 

26 

24 

36 

21 

39 

19 

17 

43 

IS 

4S 

12           48 

IO 

So 

26 

28 

20      40 

18 

42 

15 

45 

13 

47 

IO 

5° 

8  1      52 

5 

55 

28  J 

30 

5  1  7  1  6  43 

5  i4 

6  46 

5  n 

6  49 

S     9 

6  51 

S     6 

6  54 

S     3 

6  57 

5    o 

7    o 

30 

31 

15 

45 

12 

48 

9 

Si 

7 

53 

4 

$6 

i 

59 

4  S8 

2 

31 

32 

13 

47 

10 

50 

7 

53 

4 

2 

58 

4  59    7     i 

55 

5 

32 

33 

ii      49 

8 

52 

5 

55 

2 

58 

4  59 

7    i 

56 

4 

53 

7 

33 

34 

91     5i 

6 

54 

3 

57 

O 

7    o 

57 

3 

53 

7 

5° 

IO 

34 

35 

5    7  653 

5    4 

656 

5    i 

6  59 

4  58 

7    2 

4  S4 

7    6 

4  51 

7    9 

4  47 

7  IS 

35 

36 

51    55 

2 

58 

4  59 

7     i 

SS 

S 

52 

8 

48 

12 

44 

16 

36 

37 

31    57 

O 

7    o 

4 

S3 

7 

49 

ii 

45 

15 

42 

18 

37 

38 

i  !    59 

4  58 

2 

54 

6 

So 

10 

46 

H 

43        17 

39 

21 

38 

i  39 

459'  7    i 

55 

5 

9 

48 

12 

44 

16 

40 

2O 

35 

25 

39 

40 

457J7  3 

4  S3 

7    7 

4  49 

7  ii 

4  4S 

7  15 

4  4i 

7  19 

4  37 

7  23 

4  32 

7  28 

40 

41 

54       6 

5° 

IO 

46        14 

42 

18 

S8 

22 

33 

27 

29 

41 

42 

521      8 

48 

12 

43!       17 

39 

21 

SS 

2S 

3° 

3° 

25 

35 

42 

43 

49!     ii 

45 

15 

19 

36 

24 

29 

27 

33 

22 

43 

.]  44 

47!     13 

42 

1  8 

38 

22 

33 

27 

28 

32 

23 

37 

IS 

42 

44 

45 

444!  716 

4  39 

7  21 

4  35 

7  2$ 

4  3° 

7  30 

4  25 

7  35 

4  26 

7  40 

4  14 

7  46 

45 

46 

41 

19 

36 

24 

31 

29 

26 

34 

21 

39 

16 

44 

IO 

So 

46 

47 

38 

22 

33 

27 

28 

32 

23 

37 

17 

43 

12 

48 

6 

54 

47 

48 

35 

25 

30 

30 

25 

3S 

19 

13 

47 

7 

SS 

i 

59 

48 

49 

32 

28 

27 

33 

21 

39 

15 

45 

9 

51 

3 

57 

3  57 

49 

50 

429 

731 

4  23 

7  37 

4  17 

7  43 

4  ii 

7  49 

4    5 

7  SS 

3  S8 

8      2 

3  52 

8    8 

5O 

51 
52 

25 

22 

35 

38 

19 
15 

45 

13 

9 

47 

7 

2 

53 

58 

0 

3  5S 

8    o 

S 

$ 

7 

12 

47 

13 

19 

51 
52 

53 

18 

42 

ii 

49 

4 

c6 

3  57 

8    3 

5° 

IO 

43 

17 

35 

25 

53 

54 

14 

46 

7 

53 

o 

8    o 

52 

8 

4S 

IS 

37 

23 

29 

54 

55 

4   9 

75i 

4    2 

758 

3  55 

8    5 

3  47 

8  13 

3  39 

8   21 

3  3i 

8  29 

3   22 

8  38 

55 

56 

5 

55 

3  57 

8    3 

49 

ii 

19 

33 

27 

24 

36 

IS 

45 

56 

57 

o 

8  o 

52 

8 

44 

16 

35 

25 

26 

34 

17 

43 

53 

57 

58 

3  55 

S 

46 

14 

38 

22 

28 

32 

19 

9 

2   58 

9    2 

58 

59 

49 

ii 

40 

20 

31 

29 

21 

39 

ii 

49 

0 

9    o 

49 

ii 

59 

J60 

343 

817 

3  34 

8  26 

3  24 

8  36 

3  13 

8  47 

3     2 

8  58 

2  5i 

9    9 

238 

9   22 

60 

!  61 

24 

26 

34 

16 

44 

5 

55 

2  53 

9     7 

40 

20 

26 

34 

61 

62 

29 

31 

18 

42 

7 

53 

2  55 

9    5 

42 

18 

28 

32 

13 

47 

62 

63 

22 

38 

IO 

50 

258 

9    2 

44 

16 

3° 

3° 

14 

46 

i  0 

10    4 

63 

64 

13 

47 

i 

59 

47 

13 

32 

28 

16 

44 

i  S8 

IO      2 

36 

24 

64 

65 

3   3  857 

2    50 

9  10 

2  3S 

9  2S 

2    18 

9  42 

2      O 

IO      O 

i  38 

IO  22 

I     9 

10  51 

65 

66 

253 

9    7 

37 

23 

21 

39 

2 

S8 

i  39 

21 

10 

50 

o    o 

12      0 

66 

i  67 

40 

20 

23 

37 

4 

56 

i  41 

10  19 

ii 

49 

0      0 

12      0 

67 

68 

26 

34 

6 

54 

1  43 

10  17 

13 

47 

o    o 

12      O 

68 

69 

9 

51 

i  45 

10  15 

14 

46 

0      0 

12      0 

69 

0) 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

I 

1 

•M 

18° 

19° 

2O° 

21° 

22° 

23° 

24° 

nt 

| 

Declination  of  contrary  name. 

| 

98 


TABLE  XXIII. 


1 

True  Rising;  and  Setting. 

Declination  of  same  name  as  the  Latitude. 

O 

Q) 

T3 

'O 

3 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

2 

•M 

•IH 

8 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

8 

o 

/(  m 

h  m 

h  m 

h  m 

h   m 

h  m 

k  m 

h  m 

h  m 

h   m 

h   m 

h  m 

h  m 

h  m 

0     j 

0 

6   o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

6    o 

0 

2 

556 

4 

556 

4 

5  56 

4 

556 

4 

556 

4 

556 

4 

5  55 

5 

2 

4 

53 

7' 

53 

7 

52 

8 

52 

8 

9 

9 

9 

4 

6 

49 

ii 

49 

ii 

48 

12 

48 

12 

47 

47 

'3 

46 

1*4 

6 

8 

46 

14 

45 

15 

44 

16 

44 

16 

43 

17 

42 

18 

4i 

19 

8  j 

10 

542 

6  18 

)  4^ 

6  19 

5  40 

6   20 

5  39 

6   21 

538 

6   22 

5  38 

6   22 

5  37 

6  23 

10 

12 

38 

22 

37 

23 

36 

24 

35 

25 

34 

26 

33 

27 

32 

28 

12 

14 

35 

25 

33 

27 

32 

28 

29 

30 

30 

28 

32 

27 

33 

14 

16 

29 

29 

31 

28 

32 

26 

34 

25 

35 

23 

37 

22 

38 

16 

18 

27 

33 

25 

35 

24 

36 

22 

38 

20 

40 

18 

42 

17 

43 

18 

20 

523 

637 

5  21 

6  39 

5  i9 

6  41 

5  I7 

643 

5  15 

6  45 

5  i3 

6  47 

5  » 

6  49 

20 

22 

19 

41 

17 

43 

15 

45 

12 

48 

IO 

5° 

8        52 

6 

54 

22 

24 

14       46 

12 

48 

10 

50 

8 

52 

5 

55 

3       57 

o 

7    o 

24 

26 

10        50 

7 

53 

5 

55 

2 

58 

o 

7    o 

4  57 

7    3 

4  55 

5 

26 

28 

5  j      55 

3 

57 

o 

7    o 

4  57 

7    3 

4  54 

6 

9 

49 

ii 

28 

30 

5070 

458 

7    2 

4  55 

7    5 

4  52 

7    8 

448 

7  12 

4  45 

7  15 

4  42 

7  18 

30 

31 

458|           2 

55 

5 

52 

8 

49 

ii 

46 

42 

18 

39 

21 

31 

32 

55 

5 

52 

8 

49 

ii 

46 

14 

42 

$i 

39 

21 

35 

25 

32 

33 

53 

7 

49 

ii 

46 

H 

43 

17 

39 

21 

36 

24 

32 

28 

33 

34 

10 

47 

13 

43 

17 

40 

20 

36 

24 

32 

28 

28 

32 

34 

35 

447 

7  J3 

4  44 

7  16 

4  40 

7  20 

436 

7  24 

4  33 

7  27 

4  29 

7  31 

4  25 

7  35 

35  1 

36 

44 

16 

41 

19 

37 

23 

33 

27 

29 

31 

25 

35 

21 

39 

36 

37 

42 

18 

38 

22 

34 

26 

3° 

30 

26 

34 

21 

39 

17 

43 

37 

38 

39 

21 

35 

25' 

30 

30 

26 

34 

22 

38 

17 

43 

13 

47 

38 

39 

35 

25 

29 

27 

33 

23 

37 

18 

42 

13 

47 

9 

51 

39 

40 

432 

728 

428 

7  32 

4  23 

7  37 

4  19 

7  41 

4  14 

746 

4    9 

7  51 

4    4 

756 

40 

41 

29 

31 

24 

36 

20 

40 

15 

45 

10 

50 

5 

o    5S 

o 

8    o 

41 

42 

25 

35 

21 

39 

16 

44 

ii 

49 

5 

55 

o 

8    o 

3  55 

5 

42 

43 

22 

38 

17 

43 

12 

48 

7 

53 

i 

59 

3  56 

4 

50 

10 

43 

44 

iS 

42 

13 

47 

8 

52 

2 

58 

3  56 

8    4 

9 

44 

16 

44 

45 

414 

7  46 

4    9 

T  51 

4    3 

7  57 

3  57 

8    3 

3  52 

8    8 

3  45 

8  15 

3  39 

8   21 

45 

46 

10 

5 

55 

3  59 

8    i 

7 

46 

14 

40 

20 

33 

27 

46 

47 

6 

54 

0 

8    o 

54 

6 

48 

12 

19 

34 

26 

27 

33 

47 

48 

i 

59 

3  55 

5 

49 

ii 

42 

18 

35 

25 

28 

32 

20 

40 

48 

49 

357 

8    3 

10 

43 

17 

36 

24 

29 

31 

22 

38 

13 

47 

49 

50 

352 

8    8 

3  45 

8  15 

338 

8   22 

3  30 

8  30 

3  23 

837 

3    15 

845 

3    6 

854 

50 

51 

47 

13 

39 

21 

32 

28 

24 

36 

16 

44 

7 

53 

2   58 

9    2 

51 

52 

19 

33 

27 

25 

35 

17 

43 

8 

52 

2  59 

9    i 

49 

ii 

52 

53 

35 

25 

27 

33 

19 

10 

50 

0 

9    ° 

9 

40 

20 

53 

54 

29 

20 

40 

ii 

49 

2 

58 

2  52 

8 

41 

19 

30 

3° 

54 

55 

322 

838 

3  13 

847 

3    3 

857 

2  53 

9    7 

2  42 

9  18 

2  3I 

9  29 

2    18 

9  42 

55 

56 

15 

45 

55 

2  55 

9    5 

44 

16 

32 

28 

19 

41 

5 

55 

56 

57 

7 

53 

2  56 

9    4 

45 

15 

33 

27 

20 

40 

6 

54 

i  49 

IO    II 

57 

58 

258 

9    2 

47 

13 

35 

25 

22 

38 

7 

53 

i  50 

IO   IO 

3° 

30 

58 

59 

49 

ii 

36 

24 

23 

37 

8 

52 

1  51 

10    9 

31 

29 

4 

56 

59 

60 

238 

9  22 

2   25 

9  35 

2    9 

9  51 

1  52 

10    8 

1  32 

10   28 

i    5 

10  55 

0      0 

12      0 

6O 

61 

26 

34 

II 

49 

1  53 

10    7 

33 

27 

I     6 

54 

0      0 

12      0 

61 

62 

J3 

47 

i  55 

10    5 

34 

26 

6 

54 

0      0 

12     O 

62 

63 

1  S^ 

10    4 

35 

25 

7 

53 

0      0 

12      0 

63 

64 

36 

24 

52 

0      0 

12      O 

64 

65 

i    9 

10  51 

O      O!I2      O 

65 

i  66 

o   o 

12      0 

66 

!  67 

67 

1  68 

1 

68 

69 

69 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

Set. 

Ris. 

d 

CD 

•d 

+J 

24° 

25° 

26° 

27° 

28° 

29° 

3O° 

1 

I 

Declination  of  contrary  name. 

rt 

TABLE  XXIV. 


99 


Horizon-Azimuths. 

1 

Declination  of  same  name  as  the  Latitude. 

| 

% 
rt 

0?0 

o 

0.5 

1.0 

15 

2.0 

o 

2.5 

o 

3.0 

o 

3.5 

o 

4.0 

o 

4.5 

50 

55 

60 

1 

o 

0 

0 

o 

o 

0 

0 

0 

o 

o 

o 

o 

o 

,    o 

0 

O 

90.0 

89.5 

89.0 

88.5 

88.0 

87.5 

87.0 

86.5 

86.0 

85.5 

85.0 

84.5 

84.0 

0 

I0- 

O.O 

9-5 

9.0 

8.5 

8.0 

7-5 

7.0 

6.5 

5-9 

54 

4.9 

44 

3-9 

10 

O.O 

9-5 

9.0 

8.5 

7-9 

74 

O.Q 

6.4 

5-8 

5-3 

4.8 

4-3 

3-8 

15 

i  2O 

0.0 

9-5 

8.9 

8.4 

7-9 

7-3 

O.o 

6-3 

5-7 

5-2 

4-7 

4.2 

3-6 

20 

I  25 

o.o 

9-5 

8.9 

8.4 

7.8 

7.2 

6.7 

6.2 

5.6 

4-5 

4.0 

34 

25 

1  SO 

90.0 

89.4 

88.8 

88.3 

87.7 

87.1 

86.6 

86.0 

854 

84.8 

84.2 

837 

83-1 

30 

32 

o.o 

94 

8.8 

8.2 

7.6 

7.1 

6-5 

5-9 

5-3 

4-7 

4.1 

3-5 

3-° 

32 

34 

0.0 

94 

8.8 

8.2 

7.6 

7.0 

6.4 

5.8 

5.2 

4.6 

4.0 

34 

2.8 

34 

36 

o.o 

94 

8.8 

8.2 

7-5 

6.9 

6-3 

5-7 

44 

3-9 

3-2 

2.6 

36 

1  38 

o.o 

94 

8.7 

8.1 

7-5 

6.8 

6.2 

5.6 

4-9 

4-3 

3-7 

2.4 

38 

40 

90.0 

89-3 

88.7 

88.0 

87.4 

86.7 

86.1 

854 

84.8 

84.1 

83.5 

82.8 

"82.2 

40 

!  42 

o.o 

9-3 

8-7 

8.0 

7-'3 

6.6 

6.0 

5-3 

4.6 

3-9 

3-3 

2.6 

2.0 

42 

44 

o.o 

9-3 

8.6 

7-9 

7.2 

6-5 

5.8 

44 

3-7 

2.4 

1.7 

44 

>  46 

0.0 

9-3 

8.6 

7-8 

6.4 

5-7 

5.0 

4.2 

3-5 

2.S 

2.1 

1.4 

46 

48 

o.o 

9-3 

8-5 

7-8 

7.0 

6-3 

5-5 

4.8 

4.0 

3-3 

2-5 

1.8 

I.O 

48 

1  50 

90.0 

89.2 

88.5 

87.7 

86.9 

86.1 

85-3 

84.6 

83.8 

83.0 

82.2 

81.4 

80.7 

50 

51 

o.o 

9.2 

8-4 

7.6 

6.8 

6.0 

5-2 

44 

3-6 

2.8 

2.0 

1.2 

0.5 

51 

52 

0.0 

9.2 

8.4 

7.6 

6.7 

5-9 

4-3 

3-5 

2.7 

1.9 

I.O 

52 

!  53 

o.o 

9-2 

8.4 

7-5 

6.7 

5-8 

5-° 

4.2 

3-3 

2-5 

o.S 

0.0 

53 

54 

0.0 

9.1 

8-3 

7-5 

6.6 

5-7 

4-9 

4.0 

3-2 

2-3 

J-5 

0.6 

79-8 

54 

!  55 

90.0 

89.1 

88.3 

87.4 

86.5 

85.6 

84.8 

83.9 

83.0 

82.1 

81.3 

80.4 

79-5 

55 

!  56 

o.o 

9.1 

8.2 

7-3 

6.4 

5-5 

4.6 

3-7 

2.8 

1.9 

I.O 

O.I 

9.2 

56 

j  57 

0.0 

9.1 

8:2 

7-3 

6-3 

54 

4-5 

3-6 

2.6 

0.8 

79-9 

.8.9 

57 

58 

o.o 

9.1 

8.1 

7.2 

6.2 

5-3 

4-3 

34 

24 

'•5 

°-5 

9.6 

8.6 

58 

59 

0.0 

9.0 

8.1 

6.1 

4.2 

3.2 

2.2 

1.2 

o-3 

9-3 

8-3 

59 

60 

90.0 

89.0 

88.0 

87.0 

86.0 

85.0 

84.0 

83.0 

82.0 

81.0 

80.0 

79.0 

7/-9 

60 

61' 

o.o 

9.0 

7-9 

6.9 

5-9 

4.8 

3-8 

2.8 

i-7 

0.7 

79-7 

O  U 

7-5 

61 

62 

0.0 

8.9 

7-9 

6.8 

5-7 

4-7 

3-6 

2-5 

0.4 

9-3 

8.2 

62 

63 

o.o 

8-9 

7-8 

6-7 

5-5 

4.5 

34 

2-3 

1.2 

O.I 

8.9 

7.8 

6.6 

63 

64 

0.0 

8.9 

7-7 

6.6 

54 

4-3 

3-1 

2.0 

0.8 

79.7 

8-5 

74 

6.1 

64 

65.0 

90.0 

88.8 

87.6 

86.5 

85.2 

84.1 

82.9 

81.7 

80.5 

79.3 

78.1 

76.9 

75.6 

65.0 

55 

o.o 

8.8 

7-6 

6.4 

S-2 

4.0 

2.8 

1.5 

7-9 

6.6 

54 

5.5 

60 

0.0 

8.8 

7-5 

6.3 

3-9 

2.6 

1.4 

O.I 

o  Q 

7.6 

6.4 

6.O 

65 

o.o 

8.8 

7-5 

6.2 

5-° 

3-7 

2-5 

1.2 

79-9 

8.7 

74 

6.1 

1*8 

6.5 

7.O 

0.0 

8.7 

74 

6.2 

4-9 

3-6 

2-3 

I.O 

9-7 

84 

5-8 

4-5 

7.0 

67.5 

90.0 

88.7 

874 

86.1 

84.8 

83.5 

82.1 

80.8 

79-5 

78.2 

76.8 

75-5 

74-1 

67.5 

8.0 

0.0 

8-7 

7-3 

6.0 

4-7 

3-3 

2.O 

0.6 

9-3 

7.9 

6-5 

5-2 

3-8 

8.O 

8.5 

0.0 

8.6 

7-3 

5-9 

4.6 

3-2 

1.8 

0.4 

9.0 

7.6 

6.2 

4.8 

34 

8.5 

9.0 

o.o 

8.6 

7-2 

5-8 

4-5 

3-0 

1.6 

O.2 

8.8 

74 

5-9 

4-5 

9.0 

9.5 

o.o 

8.6 

5-7 

4-3 

2.8 

1.4 

0.0 

8-5 

5.6 

4.1 

2^6 

9.5 

70.0 

90.0 

88.5 

87-1 

85.6 

84.2 

82.7 

81.2 

79.7 

78.2 

76.7 

75-2 

73-7 

72.2 

70.0 

O.5 

0.0 

8.5 

7.0 

5-5 

4.0 

2-5 

I.O 

9.5 

7-9 

6.4 

4.9 

3-3 

1.8 

0.5 

1.0 

o.o 

8-5 

6-9 

54 

3-8 

2-3 

0.7 

9.2 

7.6 

6.1 

4-5 

2.9 

1.3 

1.0 

1.5 

0.0 

8.4 

6.8 

5-3 

3-7 

2.1 

0.5 

8.9 

7-3 

5-7 

4.1 

2.4 

0.8 

1.5 

2.0 

o.o 

8.4 

6.8 

3-5 

1-9 

0.2 

8.6 

5-3 

3-6 

1.9 

O.2 

2.0 

72.5 

90.0 

88.3 

86.7 

85.0 

83.3 

8l.7 

80.0 

78.3 

76.6 

74-9 

73-  r 

71.4 

69.7 

72.5 

3.O 

o.o 

8-3 

6.6 

4-9 

3-1 

1.4 

79-7 

8.0 

6.2 

4-5 

2.6 

0.9 

Q.  I 

3.0 

3.5 

0.0 

8.2 

6.5 

4.8 

2.9 

1.2 

94 

7.6 

5-8 

4.0 

2.1 

8.4 

3.5 

40 

o.o 

8.2 

6.4 

4.6 

2.7 

0.9 

9.1 

7-2 

54 

3-5 

1.6 

697 

7-7 

4.0 

4.5 

o.o 

8.1 

6.3 

44 

2-5 

0.6 

8.7 

6.8 

4-9 

2.9 

I.O 

9.0 

7.0 

4.5 

75.0 

90.0 

88.1 

86.2 

84.2 

82.3 

80.3 

78.3 

76.4 

744 

72-3 

70.3 

68.3 

66.2 

75.O 

5.5 

0.0 

8.0 

6.1 

4.0 

2.0 

0.0 

7-9 

5-9 

3-8 

1.7 

69.6 

7-5 

5-3 

5.5 

6.0 

o.o 

7-9 

6.0 

3-8 

1.7 

79.6 

7-5 

54 

3-2 

i.i 

8.9 

6.7 

44 

6.0 

6.5 

o.o 

7-9 

5-8 

3-6 

1.4 

7.0 

4.8 

2.6 

0.4 

8.1 

5.8 

34 

6.5 

7.O 

0.0 

7.8 

5.6 

34 

I.I 

8.8 

6.5 

4.2 

1.9 

69.6 

7-2 

4.8 

2-3 

7.O 

With  Declination  of  contrary  name  enter  the  Table  as  above,  but  subtract  the 

I  tabular  azimuth  from  180°.  0. 

100 


TABLE  XXIV. 


Horizon-Azimuths. 

<D 

Declination  of  same  name  as  the  Latitude. 

Q> 

'O 

'O 

•3 

£5 

ft 

I 

o 

6.0 

65 

70 

75 

So 

8.5 

90 

9.5 

o 

10.0 

105 

110 

11.5  12.0 

1 

0 

o 

o 

0 

0 

0 

o 

o 

o 

0 

o 

o 

o 

o 

o 

0 

84.0 

83.5 

83.0 

82.5 

82.0 

81.5 

Si.o 

80.5 

80.0 

79.5 

79.0 

78.5 

78.0 

0 

10 

3-9 

3-4 

2.9 

2.4 

1.9 

1.4 

0.9 

0.3 

79-9 

9-3 

8.8 

8.3 

7-8 

10 

15 

3-8 

3-3 

2.8 

2.2 

1.2 

0.7 

0.2 

9.6 

9.1 

8.6 

8.1 

7-5 

15 

20 

3-6 

3-1 

2.6 

2.0 

1.5 

0.9 

0.4 

79-9 

9-3 

8.8 

8-3 

7-7 

7.2 

20 

25 

3-4 

2.9 

2.3 

1-7 

1.2 

0.6 

O.I 

9-5 

8.9 

8-4 

7-8 

7-2 

6-7 

25 

30 

83.1 

82.5 

81.9 

8l.3 

80.7 

80.2 

79.6 

79.0 

78.5 

77.9 

77-3 

76.7 

76.1 

30 

32 

3-° 

2-3 

1.7 

1.2 

0.5 

o.o 

9.4 

8.8 

8.2 

7.6 

7.0 

6.4 

5.8 

32 

34 

2.8 

2.2 

I.c 

I.O 

79-7 

Q.2 

8-5 

7-9 

7-3 

6.7 

6.1 

5-5 

34 

36 

2.6 

2.0 

'•3 

0.7 

O.I 

9-5 

O.Q 

8.2 

7-6 

7.0 

6.4 

5-7 

5-1 

36 

38 

2.4 

1.8 

i.i 

0.5 

79-8 

9.2 

8.6 

7-9 

7-3 

6.6 

6.0 

5-3 

4-7 

38 

40 

82.2' 

81.5 

80.9 

80.2 

79-5 

78.9 

78.3 

77-6 

76.9 

76.2 

75-6 

74-9 

74-3 

40 

42 

2.0 

1.2 

0.6 

79-9 

Q.2 

8-5 

7-9 

7.2 

6.5 

5-8 

5-2 

4-4 

3-8 

42 

44 

1.7 

0.9 

o>3 

9-5 

o.Q 

8.1 

7-5 

6.7 

6.0 

5-3 

4-7 

3-9 

3-2 

44 

46 

1.4 

0.6 

79-9 

9.2 

8-5 

7-7 

7.0 

6.2 

5-5 

4-8 

4.1 

3-3 

2.6 

46 

48 

I.O 

°-3 

9-5 

8.8 

8.0 

7-2 

6-5 

5-7 

5-o 

4.2 

3-4 

2.7 

1.9 

48 

50 

80.7 

79-9 

79.1 

78.3 

77-5 

76-7 

75-9 

75.1 

74-3 

73-5 

72.7 

71.9 

71.1 

50 

51 

9.6 

8.8 

8.0 

7.2 

6.4 

5-6 

4.8 

4.0 

3-2 

2-3 

i-S 

0.7 

51 

52 

°-3 

9.4 

8.6 

7.8 

6.9 

6.1 

5-3 

4.4 

3-6 

2.8 

1.9 

i.i 

O.  I 

52 

53 

o.o 

9.2 

8-3 

7-5 

6.6 

5-8 

4-9 

4.1 

3-2 

2.4 

0.6 

69.8 

53 

54 

79.8 

8.9 

8.0 

7.2 

6-3 

5-4 

4.6 

3-7 

2.8 

1-9 

i.i 

O.2 

9-3 

54 

55 

79-5 

78.6 

77-7 

76.9 

76.0 

75-i 

74.2 

73-3 

72.4 

71.5 

70.6 

69.7 

68.8 

55 

56 

9.2 

8-3 

7-4 

6.5 

5-6 

4-7 

3-8 

2.8 

1.9 

I.O 

O.I 

9-1 

8.2 

56 

57 

8-9 

8.0 

6.1 

5-2 

4.2 

3-3 

2.3 

1.4 

0.4 

69.5 

8-5 

7-6 

57 

58 

8.6 

7-7 

£? 

5-7 

4.8 

3.8 

2.8 

1.8 

0.9 

69.9 

8.9 

7-9 

6.9 

58 

59 

8.3 

7-3 

6-3 

5-3 

4-3 

3-3 

2-3 

l-3 

°-3 

9-3 

8.3 

7-2 

6.2 

59 

6O 

77-9 

76.9 

75-9 

74-9 

73-8 

72.8 

71.8 

70.7 

69.7 

68.6 

67.6 

66.5 

65.4 

6O 

61 

7-5 

6.5 

5-4 

4.4 

3-3 

2.2 

1.2 

O.I 

9.0 

7-9 

6.9 

5-7 

4.6 

61 

62 

6.1 

4.9 

3-9 

2.7 

1.6 

0.6 

69.4 

8-3 

7-i 

6.1 

4.8 

3-7 

62 

63 

6.6 

5.6 

4.4 

3-3 

2.1 

I.O 

69.9 

8-7 

7.5 

6-3 

5-2 

3-9 

2.8 

63 

64 

6.1 

5.0 

3.8 

2.7 

l'S 

0.3 

9.1 

7-9 

6.7 

5-4 

4-3 

2.9 

1-7 

64 

65.  0 

75.6 

74-5 

73-2 

72.0 

70.7 

69.5 

68.3 

67.0 

65.8 

64.4 

63.2 

61.8 

60.5 

65.0 

5.5 

5-4 

4.2 

2.9 

1.7 

0.4 

9.1 

7.8 

6.5 

5-3 

3-9 

2.6 

1.3 

59-9 

5.5 

6.0 

5-1 

3-8 

2.6 

1.3 

0.0 

8.7 

7-4 

6.1 

4-7 

3-4 

2.0 

0.7 

9-3 

6.0 

6.5 

4.8 

3-5 

2.2 

0.9 

69.6 

8.2 

6.9 

5.6 

4.2 

2.8 

1.4 

o.o 

8.6 

6.5 

7.0 

4.5 

3-2 

1.8 

9.1 

7.8 

6.4 

3-6 

2.2 

0.8 

59.3 

7-9 

7.0 

67.5 

74.1 

72.8 

71.4 

70.1 

68.7 

67.3 

65.9 

64.5 

63.0 

6l.6 

60.  i 

58.6 

57-1 

67.5 

80 

3-8 

2.4 

I.O 

69.6 

8.2 

6.8 

5-3 

3-9 

2.4 

0.9 

59-4 

7.8 

6-3 

8.0 

8.5 

34 

2.0 

0.6 

9-i 

7-7 

6.2 

4-7 

3-2 

1.7 

O.2 

8.6 

7.0 

5-4 

8.5 

9.0 

3-9 

1.6 

O.I 

8.6 

7.2 

5-6 

4.1 

2.6 

I.O 

59-4 

7.8 

6.2 

4-5 

9.0 

9.5 

2.6 

i.i 

69.6 

8.1 

6.6 

3-5 

1.9 

°-3 

8.6 

7.0 

5.3 

3-6 

9.5 

70.0 

72.2 

70.7 

69.1 

67.6 

66.0 

64.4 

62.8 

61.2 

59-5 

57.8 

56.1 

54.3 

52.6 

70.0 

0.5 

1.8 

O.2 

8.6 

7.0 

5-4 

3-7 

2.1 

0.4 

8.7 

6.9 

5-1 

3-3 

1.5 

O.5  ! 

10 

'•3 

69.7 

8.0 

6.4 

4-7 

3-° 

i-3 

59-5 

7-8 

6.0 

4.1 

2.2 

o-3 

1.0 

1.5 

0.8 

9.1 

7-4 

5-7 

4.0 

2.2 

0.5 

8.6 

6.8 

5'° 

3-° 

I.I 

49.1 

1.5 

2.0 

O.2 

8.5 

6.8 

3-2 

1.4 

59-6 

7-7 

5-8 

3-9 

1.9 

49.8 

7-7 

2.0 

72.5 

69-7 

67.9 

66.1 

64-3 

62.4 

60.5 

58.6 

56.7 

54-7 

52-7 

50.6 

48.5 

46.3 

72.5 

3.0 

9.1 

7.2 

5-4 

3-5 

1.6 

59-6 

7.6 

5.6 

3-5 

1.4 

49.2 

7.0 

4-7 

3.O 

3.5 

8.4 

6-5 

4.6 

2.6 

0.7 

8.6 

6.6 

4-5 

2-3 

O.I 

7.8 

5-4 

3.0 

3.5 

4.0 

7-7 

5-7 

3-8 

1.7 

59-7 

7-5 

5-4 

3-2 

0.9 

48.6 

6.2 

3-7 

i.i 

4.0 

4.5 

7.0 

4-9 

2.9 

0.7 

8.6 

6.4 

4.2 

1.8 

49-5 

7.0 

4-4 

1.8 

38.9 

4.5 

75.0 
5.5 

66.2 
5-3 

64.0 
3-1 

61.9 
0.9 

'I! 

57-5 

6.2 

55-2 
3-8 

52.8 
'•3 

5°-4 
48.8 

47-9 
6.1 

45-2 
3-3 

42.5 
o-3 

39-6 

7-2 

36-5 
3-8 

75.0  ! 
3.5 

6.O 

4-4 

2.1 

59-8 

7-4 

4.9 

2-3 

49-7 

7.0 

4.1 

i.i 

37-9 

4-5 

0.7 

6.O 

6.5 

3-4 

I.O 

8.6 

6.0 

3-4 

0.7 

7-9 

5-° 

1.9 

38.7 

5-2 

1.3 

27.0 

6.5  | 

7.0 

2-3 

59.8 

7.2 

4-5 

1.8 

48.9 

5-9 

2.8 

39-5 

5-9 

2.O 

27.6 

22.4 

7.0  i 

With  Declination  of  contrary  name  enter  the  Table  as  above,  but  subtract  the  | 

tabular  azimuth  from  180°.0. 

TABLE  XXIV. 


101 


Horizon-Azimuths. 

1 

Declination  of  same  name  as  the  Latitude. 

| 

;  1 

« 

3 

12.O 

0 

12.5 

o 

13.O 

o 

13.5 

14.0  14.5 

1.5.0 

15.5 

16.0 

16.5 

170 

175 

0 

18.0 

1 

a 

0 

o" 

o 

o 

0 

o 

o 

o 

o 

o 

0 

0 

0 

o 

o 

78.0 

77-5 

77.0    76.5 

76.0 

75-5 

75.0 

74-5 

74.0 

73-5 

73.O 

72.5 

72.0 

O 

10 

77.8 

7-3 

6.8 

6.3 

5.8 

5-3 

4-7 

4.2 

3-7 

3-2 

2.7 

2.2 

10 

15 

7-5 

7-1 

6.5 

6.0 

5-5 

44 

3-9 

34 

2.9 

2-3 

1.8 

1.3 

15 

2O 

7-2 

6.7 

6.2 

5.6 

4-5 

4.0 

3-5 

2.9 

2.4 

1.9 

1.3 

0.8 

20      1 

25 

6.7 

6.2 

5.6 

4-5 

3-9 

34 

2.9 

2-3 

1.7 

1.2 

0.6 

O.I 

25 

3O 

76, 

75-5 

75-° 

744 

73-8 

73-2 

72.6 

72.0 

71.4 

70.8 

70.3 

69.7 

69.1 

30 

32 

5-8 

5-2 

4-7 

4.0 

34 

2.8 

2.2 

1.6 

l.O 

0.4 

69.8 

9-2 

8.6 

32 

34 

5-5 

4.9 

4-3 

3-6 

3-° 

2.4 

1.8 

1.2 

°-5 

0.0 

9-3 

8.7 

8.1 

34 

36 

5-1 

4-5 

3-9 

3-2 

2.6 

2.O 

1.3 

0.7 

o.o 

69.5 

8.8 

8.2 

7-5 

36 

3§ 

4-7 

4.0 

34 

2.8 

2.1 

J-5 

0.8 

O.2 

69.5 

8.9 

8.2 

7.6 

6.9 

38 

4O 

74-3 

73-6 

72.9 

72.2 

71.6 

70.9 

70-3 

69.6 

68.9 

68.2 

67.6 

66.9 

66.2 

40 

41 

4.0 

3-3 

2-7 

2.O 

*-3 

0.6 

0.0 

9.2 

8.6 

7-9 

7.2 

6.5 

5-8 

41 

42 

3-8 

3-1 

2.4 

1.7 

1.0 

0-3 

69.6 

8.9 

8.2 

7-5 

6.8 

6.1 

54 

42 

43 

3-5 

2.8 

2.1 

1.4 

0.7 

o.o 

9.3 

8.6 

7-8 

7-1 

6.4 

5-7 

5-° 

43 

44 

3-2 

2-5 

1.8 

I.I 

0.4 

69.6 

8.9 

8.2 

74 

6.7 

6.0 

5-3 

4.6 

44 

!  45 

72.9 

72.2 

7i.5 

70.7 

70.0 

69.3 

68.5 

67.8 

67.0 

66.3 

65.6 

64.8 

64.1 

45 

46 

2.6 

1.8 

i.i 

0.4 

69.6 

8-9 

8.1 

74 

6.6 

5-9 

C.I 

4-3 

3-6 

46 

47 

2-3 

1  5 

0.7 

0.0 

9.2 

8-5 

7-7 

6.9 

6.2 

54 

4.6 

3-8 

3-1 

47 

48 

1.9 

i.i 

69.6 

8.8 

8.0 

7-2 

6.4 

5-7 

4.9 

4.1 

3-3 

2-5 

48 

49 

f-S 

0.7 

69.9 

9-2 

8.4 

7.6 

6.8 

5-9 

4-3 

3-5 

2.7 

1.9 

49 

5O 

71.1 

70.3 

69.5 

68.7 

67.9 

67-1 

66.3 

654 

64.6 

63.8 

63.0 

62.1 

61-3 

50 

51 

0.7 

69.9 

9.1 

8.2 

74 

6.5 

5-7 

4.9 

4.0 

3-2 

2.4 

1-5 

0.6 

51 

52 

0.3 

94 

8.6 

7.7 

6.9 

6.0 

4-3 

34 

2.5 

0.8 

59-9 

52 

53 

69.8 

8.9 

8.1 

7.2 

6-3 

54 

4-5 

3-6 

2.7 

1.8 

1.0 

o.o 

9.1 

53 

54 

9-3 

8.4 

7-5 

6.6 

5-7 

4.8 

3-9 

2.9 

2.0 

i.i 

0.2 

59-2 

8-3 

54 

55 

68.8 

67.8 

66.9 

66.0 

65.1 

64.1 

63-2 

62.2 

6l.3 

60.3 

594 

584 

574 

55 

56 

57 

8.2 

7.6 

7-2 

6.6 

6-3 
5-6 

5-3 
4.6 

4.4 
3-6 

34 

2.6 

24 
1.6 

1.4 

0.6 

0.5 

sll 

8-5 
7-5 

n 

6.4 
54 

56 
57 

58 

6.9 

5-9 

4.9 

3-9 

2.8 

1.8 

0.8 

59-7 

87 

7.6 

6.5 

54 

4-3 

58 

59 

6.2 

5-2 

4.1 

2.0 

0.9 

59-8 

8-7 

77 

6.5 

54 

4-3 

59 

60 

654 

64.4 

63-3 

62.2 

61.1 

59-9 

58.8 

57-7 

56.6 

554 

54-2 

53.1 

51.8 

69 

61 

4.6 

3-5 

2.4 

1.2 

O.I 

•8.9 

7-8 

6-5 

54 

4.2 

2.9 

1.7 

0.4 

61 

62 

3-7 

2-5 

1.4 

O.2 

59-0 

7-8 

6.6 

5-3 

4.1 

2.8 

0.2 

48.8 

62 

63 

2.8 

o-3 

59-o 

7-8 

6.5 

5-3 

3-9 

2.6 

1.3 

49-9 

48.5 

7-1 

63 

64 

i-7 

0.4 

7.8 

6.5 

5-2 

3-8 

2.4 

l.O 

49.6 

8.2 

6.7 

5-2 

64 

65.O 

60.5 

59-2 

57-8 

56.5 

55-i 

53-7 

52.2 

50.8 

49.3 

47-8 

46.2 

44.6 

43-o 

65.0 

5.5 

59-9 

8-5 

7-1 

5-7 

4-3 

2.9 

1.4 

49-9 

8.4 

6.8 

5-2 

•3-5 

1.8 

5.5 

6.O 

9.3 

7-8 

6.4 

5-0 

3-5 

2.0 

0.5 

8.9 

7-3 

5-7 

4.1 

2-3 

0.6 

6.0 

6.5 

8.6 

7.1 

5-7 

4.2 

2-7 

I.I 

49-5 

7-9 

6.2 

4.6 

2.9 

i.i 

39-2 

6.5 

7.0 

7-9 

6.4 

4.9 

3-3 

1.8 

0.2 

8-5 

6.8 

5-1 

34 

!.6 

39-7 

7-7 

7.0 

167.5 

57-1 

55-6 

54«° 

524 

50.8 

49-2 

474 

45-7 

43-9 

42.1 

40.2 

38.2 

36.1 

67.5 

8.O 

6.3 

4-7 

3.1 

1.4 

49.8 

8.1 

6-3 

4-5 

2.6 

0.7 

38.7 

6.6 

44 

8.0 

'    8.5 

54 

3.8 

2.1 

0.4 

8-7 

6.9 

5.1 

3-2 

1.2 

39-2 

7-1 

4.9 

2-5 

8.5 

9O 

4-5 

2.8 

I.I 

49-3 

7-5 

5-7 

3-8 

1.8 

39-7 

7.6 

54 

0.4 

9.0 

9.5 

3-6 

1.8 

o.o 

8.2 

6-3 

44 

2.4 

0-3 

5-8 

3-5 

0.9 

28.1 

9.5 

7O.O 

52.6 

50.7 

48.9 

47.0 

45-o 

43-° 

40.8 

38.6 

36.3 

33-9 

3r-3 

28.5 

254 

70.O 

O.5 

1.5 

49.6 

7.6 

5.6 

3.6 

1.4 

39-2 

6.8 

4-3 

28.9 

5-7 

2.2 

0.5 

l.O 

o-3 

8-3. 

6-3 

4.2 

2.0 

39-7 

74 

4.8 

2.1 

29-3 

6.1 

2.5 

I8.3 

1.0 

1.5 

49.1 

7-o 

4-9 

2.6 

0-3 

7-9 

54 

2.6 

29-7 

6.5 

2.9 

18.6 

'3-1 

1.5 

2.O 

7-7 

5.6 

3-3 

0.9 

38.5 

5-9 

O.I 

6.9 

3-2 

18.9 

13-3 

o.o 

2.0 

72.5 

46.3 

44.0 

41.6 

39.1 

364 

33-6 

30.6 

27.3 

23.6 

19.1 

13-5 

o.o 

72.5 

3.O 

4-7 

2-3 

39-7 

7.0 

4.1 

27.8 

3-9 

194 

13-7 

o.o 

3.O 

3.5 

3-° 

0.4 

7-7 

4-7 

1.6 

28^2 

44 

19.7 

13-9 

0.0 

3.5 

4.0 

i.i 

38.3 

5-3 

2.1 

28.6 

4-7 

O.2 

14.1 

OiO 

4.0 

4.5 

38-9       5-9 

2.7 

29.1 

5.1 

o-5 

14-5 

0.0 

4.5 

i 

"With  Declination  of  contrary  name  enter  the  Table  as  above,  but  subtract  the 

tabular  azimuth  from  180°.0. 

102 


TABLE  XXIV. 


Horizon-  Azimuths. 

1 

Declination  of  same  name  as  the  Latitude. 

6 

1 

o 

18.0 

o 

18.5 

19.0 

19.5 

20.0 

20.5 

o 

21.0 

o 

21.5 

o 

22.0 

22.5 

23.0 

23.5 

24.O 

I) 

1 

o 

0 

0 

0 

0 

0 

0 

o 

0 

0 

0 

0 

0 

o 

0 

0 

72.0 

71-5 

71.0 

7°-5 

70.0 

69.5 

69.0 

68.5 

68.0 

67.5 

67.0 

66.5 

66.0 

O 

10 

1.2 

0.7 

O.2 

69.7 

9.2 

8.7 

8.2 

7-7 

7-1 

6.6 

6.1 

5.6 

10 

15 

1.3 

0.8 

69.8 

9-3 

8.7 

8.2 

7-7 

7.2 

6.7 

6.1 

5.6 

5-1 

15 

20 

0.8 

0.3 

69J 

9.2 

8.6 

8.1 

7.6 

7.0 

6.5 

6.0 

54 

4.9 

4-3 

20 

25 

O.I 

69.5 

8.9 

8.4 

7.8 

7-3 

6.7 

6.1 

5.6 

5-° 

4-5 

3-9 

3-3 

25 

30 

69.1 

68.5 

67.9 

67-3 

66.7 

66.2 

65-6 

65.0 

64.4 

64.8 

63-2 

62.6 

62.0 

30 

32 

8.6 

8.0 

74 

6.8 

6.2 

5.6 

5.O 

44 

3-8 

3-2 

2.6 

2.O 

i-3 

32 

34 

8.1 

7-5 

6.9 

6.2 

5.6 

5-° 

44 

3-8 

3-1 

2-5 

1.9 

1-3 

0.6 

34 

36 

7-5 

6.9 

6.3 

5-6 

4-3 

3-7 

1.8 

i.i 

0.5 

59-8 

36 

38 

6.9 

6.2 

5-6 

4.9 

4-3 

3-6 

3-° 

2-3 

1.6 

0.9 

°-3 

59-6 

8-9 

38 

4O 

66.2 

65.6 

64.9 

64.2 

63-5 

62.8 

62.1 

61.4 

60.7 

60.0 

59-3 

58.7 

57-9 

4O 

41 

42 

5.8 
54 

5-2 

4-7 

4-5 
4.0 

3-8 
3-3 

& 

2-3 
1.9 

1-7 

1.2 

0.9 
0.4 

O.2 

59-7 

59-5 
9.0 

8^ 

8.2 

7.6 

74 
6.8 

41 
42 

43 

5-° 

4-3 

3-6 

2.8 

2.1 

1.4 

0.7 

59-9 

9.2 

8.4 

7-7 

7.0 

6.2 

43 

44 

4.6 

3-8 

3-1 

2-3 

1.6 

0.9 

O.2 

94 

8.6 

7.8 

6.4 

5.6 

44 

45 

64.1 

63.3 

62.6 

61.8 

61.1 

60.3 

59-6 

58.8 

58.0 

57-2 

56.5 

55-7 

54-9 

45 

46 

3.6 

2.8 

2.1 

J-3 

o>5 

59-7 

9-0- 

8.2 

74 

6.6 

5-8 

5>o 

4.2 

46 

47 

3-1 

2-3 

I-S 

0.7 

59-9 

9.1 

8-3 

7-5 

6.7 

5-9 

4-3 

34 

47 

48 

2-5 

P-7 

0.9 

O.I 

9-3 

8.4 

7.6 

6.8 

6.0 

4-3 

3-5 

2.6 

48 

49 

1.9 

i.i 

o-3 

594 

8.6 

7-7 

6.9 

6.0 

5-2 

4-3 

3-5 

2.6 

J-7 

49 

50 

61.3 

60.4 

59-6 

58.7 

57-9 

57-° 

56.1 

55-2 

544 

53-5 

52.6 

5!-7 

50.8 

50 

51 

0.6 

59-7 

8.9 

8.0 

6.2 

5-3 

44 

3-5 

2.6 

1.6 

0.7 

49.8 

51 

52 

59-9 

9.0 

8.1 

7-2 

£3 

5-3 

44 

3-5 

2-5 

1.6 

0.6 

49-7 

8.7 

52 

53 

9-i 

8.2 

7-3 

6.3 

54 

44 

34 

2-5 

0-5 

49-5 

8.6 

7-5 

53 

54 

8-3 

7-3 

6.4 

54 

44 

34 

2.4 

1.4 

0.4 

494 

8-3 

74 

6.2 

54 

55 

574 

56.4 

554 

544 

534 

524 

51.3 

50-3 

49-2 

48.1 

47.1 

46.0 

44-8 

55 

56 

6.4 

54 

44 

3-3 

2-3 

1.2 

0.2 

49.0 

7-9 

6.8 

5-7 

4.6 

3-3 

56 

57 

54 

44 

3-3 

2.2 

i.i 

O.O 

48.9 

7-7 

6-5 

54 

4.2 

3.0 

i-7 

57 

58 

4-3 

2.1 

0.9 

49-8 

48.6 

7-5 

6.2 

3-8 

2.5 

1.2 

39-9 

58 

59 

2.0 

0.8 

49.6 

8.4 

7.2 

5-9 

4.6 

3-3 

2.0 

0.7 

39-3 

7.8 

59 

60.0 

51.8 

50.6 

494 

48.1 

46.8 

45-5 

44-2 

42.8 

41.4 

4O.I 

38.6 

3M 

35-6 

60.0 

0.5 

i.i 

49.9 

8.6 

7-3 

6.0 

4.6 

3-3 

1.9 

0.4 

39-o 

7-5 

5-9 

4-3 

O5 

l.O 

0.4 

7.8 

6.5 

5-1 

3-7 

2-3 

0.9 

394 

7-9 

4-7 

3.0 

10 

1.5 

49.6 

8'3 

7.0 

5-6 

4.2 

2.7 

39-8 

6-7 

0 

3-3 

1.5 

1.5  i 

2.0 

8.8 

7-5 

6.1 

4-7 

3-2 

O.2 

8.7 

7-1 

54 

3-7 

1.9 

o.o 

2.0 

62.5 

48.0 

46.6 

45-  * 

43-7 

42.2 

40.6 

29.0 

374 

35-8 

34.0 

32.2 

30-3 

28.3 

62.5 

3.0 

5-7 

4.1 

2.6 

39-5 

7.8 

6.1 

44 

2-5 

0.6 

28.6 

6.4 

3.O 

3.5 

6.2 

4-7 

3-1 

1.5 

39-9 

8-3 

6.5 

4-7 

2.9 

0.9 

28.9 

6.6 

4-3 

3.5 

4.0 

5'2 

3-6 

2.0 

o«3 

8-7 

7.0 

e  i 

3-3 

29.2 

7.0 

4-5 

1.9 

4.0 

4.5 

4.1 

2-5 

0.8 

39-1 

74 

5-5 

3'-6 

1.6 

29-5 

7.2 

4.8 

2.2 

19.1 

4.5 

65.0 

43.0 

41-3 

39-6 

37-8 

36.0 

34-o 

32.0 

29.8 

27-5 

25.1 

22.4 

194 

15.6 

65.0 

5.5 

1.8 

0.0 

8.2 

6.4 

44 

2.4 

0.2 

7.8 

5-3 

2.7 

19.6 

15-9 

n.  i 

5.5 

60 

0.6 

38.7 

6.8 

4.9 

2.7 

0.6 

28.2 

5-6 

2.9 

19.8 

16.2 

II.4 

o.o 

6.O 

65 

39-2 

7-3 

5-3 

3-2 

0.9 

28.6 

6.0 

3-2 

o.o 

16.3 

1  1.6 

O.O 

6.5 

7.0 

7-7 

5-7 

3.6 

28.9 

6-3 

3-5 

0.2 

16.5 

11.7 

0.0 

7.0 

67.5 

36.1 

34-o 

31.7 

29-3 

26.6 

23-8 

20.5 

I6.7 

11.8 

0.0 

67.5 

8.O 

44 

2.1 

29.7 

7.0 

4.1 

0.8 

17.0 

II-9 

o.o 

8.O 

8.5 

2-5 

o.o 

74 

44 

i.i 

17.2 

I2.I 

O.O 

8.5 

9.0 

0.4 

27.7 

4-7 

17-3 

12.3 

O.O 

9.O 

9.5 

28.1 

1.6 

17.6 

12.4 

0.0 

9.5 

70.0 

254 

21.9 

17.8 

12.6 

o.o 

70.0 

0.5 

2.2 

18.1 

12.8 

0.0 

0.5  ! 

1.0 

I8.3 

12.9 

0.0 

10  i 

1.5 

!3-! 

o.o 

1.5 

2.0 

0.0 

2O 

"With  Declination  of  contrary  name  enter  the  Table  as  above,  but  subtract  the 

tabular  azimuth  from  180  .0. 

TABLE  XXIV. 


103 


Horizou-  Azimuths. 

0) 

Declination  of  same  name  as  the  Latitude. 

| 

1 

24.O 

245 

25.0 

255 

26.O  26.5 

27.0 

27.5 

o 

28.O 

28.5 

29.0 

29.5 

0 

30.0 

2 

H) 

o 

o 

0 

o 

O 

o 

o 

0 

0 

0 

0 

o 

O 

o 

o 

0 

66.0 

65.5 

65.0 

64.5 

64.0 

63'5 

63.0 

62.5 

62.0 

61.5 

6  i.o 

60.5 

6O.O 

0 

4 

5-9 

5-4 

4-9 

4-4 

3-9 

3-4 

2-9 

2.4 

1.9 

1.4 

0.9 

0.4 

59-9 

4 

8 

5-7 

5-2 

4-7 

4.2 

3-7 

3-2 

2-7 

2.2 

1.7 

1.2 

0.7 

O.2 

9-7 

8 

12 

5-4 

4-9 

4.4 

3-9 

3-4 

2.9 

2.4 

1.9 

1.3 

0.8 

0.3 

59-8 

9-3 

12 

16 

4-4 

3-9 

3-4 

2.9 

2.4 

1.8 

I-3 

0.8 

O.2 

59-7 

9-2 

8-7 

16 

20 

64-3 

63.8 

63-3 

62.7 

62.2 

.61.7 

61.1 

60.6 

60.0 

59-5 

58.9 

58.4 

57-9 

20 

22 

4.0 

3-4 

2.9 

2-3 

1.8 

1.2 

0.7 

O.I 

59-6 

9.0 

8.5 

7-9 

7-4 

22 

24 

3-6 

3-° 

2.4 

1.9 

1.3 

o.S 

O.2 

59-6 

9-i 

8-5 

8.0 

7-4 

6.8 

24 

26 

3-1  i    2.5 

1.9 

1.4 

0.8 

0.3 

59-7 

9.1 

8.5 

7-9 

7-4 

6.8 

6.2 

26 

28 

2.6 

2.O 

1.4 

0.8 

0.2 

59-7 

9.1 

8.5 

7-9 

7-3 

6-7 

6.1 

5-5 

28 

30 

62.0 

6l.4 

60.8 

60.2 

59-6 

59.0 

•58.4 

57.8 

57-2 

56.6 

56.0 

55-3 

54-7 

30 

31 

i.j 

I.I 

°-5 

59-9 

9.2 

o  O 

8.0 

7-4 

6.8 

6.2 

5-5 

•  4-9 

4-3 

31 

32 

1.3 

0.7 

O.I 

9-5 

8.9 

8-3 

7.6 

7.0 

6.4 

5.8 

4-5 

3-9 

32 

33 

I.O 

0.4 

59-8 

9.1 

8.5 

7-9 

7-2 

6.6 

6.0 

5-3 

4-7 

4.0 

3-4 

33 

34 

0.6 

O.O 

9-4 

8.7 

8.1 

7-4 

6.8 

6.2 

5-5 

4-9 

4.2 

3-6 

2.9 

34 

35 

60.2 

59-6 

58.9 

58.3 

57-7 

57-0 

56.4 

55-7 

55-o 

54-4 

53-7 

53-i  !  52-4 

35 

36 

59.8 

9.2 

8.5 

7-9 

7.2 

6.5 

5-9 

S-2 

4-5 

3-9 

3.2 

2.5       1.8 

33 

37 

9.4 

8.7 

8.1 

7-4 

6.7 

6.0 

5-4 

4-7 

4.0 

3-3       2.6 

1.9 

1.2 

37 

'  38  . 

8.9       8.3 

7.6 

6.9 

6.2 

5-5 

4.8 

4.1 

3-4 

2.7    j      2.0 

'•3 

0.6 

33 

39 

8.4       7-8 

6.4 

5-7 

4-3 

3-5 

2.8 

2.1          1.4 

0.7  |    o.o 

39 

40 

57-9     57-2 

56.5     55-8 

55-1 

54-4 

53-7 

52-9 

52.2     51.5  |  50.7 

50.0  !  49.3 

40 

41 

7-4 

6.7 

5-9 

5-2 

4-5 

3-8 

3-° 

2.3 

1.5 

0.8       o.o 

49.3     8.5 

41 

42 

6.8 

6.1 

5-3 

4-6 

3-9 

2.3 

1.6 

0.8 

O.I 

49-3 

8-5       7-7 

42 

43 
44 

6.2 

5-6 

5-5 
4.8 

4-7 
4.0 

3-9 
3-2 

3-2 

2-5 

2.4 

1.6 
0.9 

0.8 
o.o 

O.I 

49-3 

49-3 
8.4 

8.5 
7.6 

7-7       6.9 
6.8       6.0 

43 
44 

i  45 

54-9 

54-1 

53-3 

52.5 

51.7 

5°-9 

50.1 

49-2 

48.4 

47-5 

46.7 

45-9     45  -° 

45 

1  46 

4.2 

3-4 

2-5 

0.9 

o.o 

49.2 

8-3 

7-5 

6.6 

5-7 

4.9       4.0 

46 

47 

3-4 

2.6 

1-7 

0.9 

0.0 

49.1 

8-3 

7-4 

6.5 

5-6 

4-7 

3.8  ;     2.9 

47 

48 

2.6 

1.7 

0.8 

o.o 

49.1 

8.2 

7-3 

6.4 

5-4 

4-5 

3-6 

2.6    ;       1-7 

48 

49 

J-7 

0.8 

49.9 

49.0 

8.1 

7-2 

6.2 

5-3 

4-3 

3-3 

2.4 

1.4    !       0.4 

49 

50 

50.8     49.8 

48.9 

48.0 

47.0 

46.1 

45-  i 

44.1 

43-i 

42-1 

41.1 

4O.O     i     38.9 

50 

51 

49.8  1    8.8 

7.8 

6.8 

5-9 

4-9 

-    3-8 

2.8 

1.8 

0.7 

39-6 

38.5  !     7-4 

51 

52 

8.7       7.7 

6.7 

5-6 

4.6 

3.6 

2-5 

1.4 

o-3 

39-2 

8.0 

6.9       5.7 

52 

53 

7-5 

6.5 

5-4 

4-3 

3-3 

2.2 

I.O 

39-9 

38.7 

7-5 

6-3 

5.i  ;    3-8 

53 

54 

6.2 

5-1. 

4.0 

2.9 

1.8 

0.6 

39-4 

8.2 

7.0 

5-7 

4-4 

54 

55.0 

41.8 

43-7  !  42.5 

41.4 

40.2 

38.9 

37-7 

36-4 

35-i 

33-7 

32-3 

30.9     29.3 

55.0 

5.5 

4.1 

2.9 

1.7 

0.5 

39-3 

8.0 

6-7 

5-4 

4.0 

2.6 

i.i 

29.6 

8.0 

5.5 

6.0 

3-3 

2.1 

0.9 

8-4 

7.1 

5-7 

4-3 

2-9 

1.4 

29.9 

8-3 

6.6 

6.0 

65 

2.5           1.2 

o.o 

31:7 

7-4 

6.1 

4.6 

3-2 

O.I 

8-5 

6.8 

5-° 

6.5 

7.0 

1.7          0.4 

39-1 

7-8 

6-4 

5-° 

3-5 

2.O 

o-5 

28.8 

7-1 

5-3 

3-4 

7.0 

57.5 

3.0 

40.8 

39-9 

39-5 
8-5 

38.1 

36.8 

5-7 

35-3 
4.2 

33-8 

2.6 

32-3 

3°-7 
29.4 

29.1 
7.6 

27.4 
5-8 

3 

23.6 

21.5 
19-3 

57.5 

8.0 

85 

8.9 

7-5 

6!o 

4-5 

3-° 

1.4 

29.7 

7-9 

6.1 

4.0 

1.9 

19.6 

16.9 

8.5 

9.0 

7.8       6.4 

4-9 

3-3 

o.o 

8.2 

6-3 

4-3 

2.1 

19.7 

17.0 

13.8 

9.O 

9.5 

6.7       5.2 

3-6 

2.O 

°-3 

28.5 

6.6 

4-5 

2-3 

19.9 

17.2 

14,1 

-9.9 

9.5 

600 

35-6     34-0 

32.3 

30.6 

28.8 

26.8 

24.8 

2.6 

20.  r 

17.4 

14.2 

IO.O 

o.o 

60.0 

0.5 

4-3 

2.6 

0.9 

29.0 

7.1 

5° 

2.8 

0.4 

17.6 

14.2 

IO.I 

o.o 

0.5 

1.0 

1.2 

29-3 

7-4 

5-3 

o-5 

17.7 

14-5 

10.2 

0.0 

1.0 

1.5 

1.5 

29.7 

7-7 

5.6 

3-3 

pj 

18.0 

14.6 

10.3 

0.0 

1.5 

2.O 

o.o 

8.0 

5.8 

3-5 

I.O 

18.1 

14.8 

10.4 

o.o 

2.0 

62.5 

28.3 

26.1 

23-8 

21.2 

18.3 

14.9 

10.5 

0.0 

62.5 

3.O 

6-4 

4.0 

1.4 

I8.5 

IS-1 

10.6 

o.o 

3.O 

3.5 

4-3 

18.7 

15.2 

10.7 

o.o 

3.5 

4.0 

1.9 

18.9 

10.8 

o.o 

4.0 

4.5 

19.1 

15.6 

I  I.O 

o.o 

. 

4.5 

With  Declination  of  contrary  name  enter  the  Table  as  above,  but  subtract  the 

tabular  azimuth  from  180°.0. 

104 


TABLE  XXV. 


Position-  Angles  for  Horizon-Azimuths. 

Lat. 

Declination  of  same  or  contrary  name. 

O° 

5° 

10° 

12° 

14° 

16° 

18° 

20° 

22° 

24° 

26° 

28° 

3O° 

Lat. 

0 

0 

0 

o 

o 

o 

o 

o 

o 

o 

o 

0 

0 

0 

o 

0 

90.0 

90.0 

90.0 

90.0 

90.0 

90.0 

90.0 

90.0 

9c 

.0 

9 

3.O 

90.0 

90.0 

90.0 

0 

2 

88.0 

88.0 

88.0 

88.0 

87.9 

87.9 

87.9 

87. 

-) 

87.8 

8 

7.8 

87.8 

87.7 

87.7 

2 

4 

6.0 

6.0 

6.0 

5-9 

5-9 

5-8 

5.8 

5- 

j 

5 

•7 

5.6 

5-6 

5-5 

5-4 

4 

6 

4.0 

4.0 

3-9 

3-9 

3-8 

3-8 

3-7 

3-6 

3-5 

3-4 

3-3 

3-2 

6 

8 

2.0 

2.0 

•9 

1.8 

1.8 

1.6 

i. 

5 

i 

•4 

1.2 

i.i 

0.9 

0.8 

8 

10 

8o.O 

80.0 

79-9 

79.8 

79-7 

79.6 

79-5 

79-4 

79.2 

79.0 

78.9 

78.6 

78.4 

10 

12 

78.0 

77-9 

7-8 

7-7 

7.6 

7-5 

7-4 

7.2 

7.0 

6.8 

6.6 

6.4 

6.1 

12 

14 

6.0 

5-9 

i 

0 

.0 

5-7 

5-6 

5-4 

5-3 

5- 

i 

4-9 

4.6 

4-4 

4.1 

3-8 

14 

16 

4.0 

3-9 

3-8 

3-6 

3-5 

3-3 

3-1 

3-° 

2.7 

2.4 

2.1 

1.8 

1.4 

16 

18 

2.O 

1.9 

] 

•7 

1.6 

1.4 

1.2 

I.O 

0.8 

c 

•5 

0.2 

69.9 

69-5 

69.1 

18 

20 

7O.O 

69.9 

69.7 

69.5 

694 

69.2 

68.9 

68.7 

68.4 

68.0 

67.6 

67.2 

66.7 

20 

22 

68.0 

7-9 

7-7 

7-5 

7-3 

7.1 

6.8 

6.5 

6.2 

5-8 

54 

4.9 

4.4 

22 

24 

6.0 

5-9 

i 

.6 

5-4 

5-2 

5'0 

4.7 

4.4 

4.0 

3-6 

3-1 

2.6 

2.0 

24 

26 

4.0 

3-9 

3-6 

3-4 

3-1 

2.9 

2-5 

2.2 

.8 

1.3 

0.8 

O.2 

59-6 

26 

28 

2.0 

1.9 

] 

•5 

i.i 

0.8 

0.4 

O.O 

59.6 

59-1 

58.5 

57-9 

7.2 

28 

30 

60.0 

59-9 

59-5 

59-2 

59-0 

58.6 

58.3 

57.8 

57-4 

56.8 

56.2 

55-5 

54-7 

3O  ' 

32 

58.0 

7-9 

7-5 

7.2 

6.9 

6.5 

6.1 

5- 

7 

5 

•  *• 

4-5 

3-9 

2.2 

32  j 

34 

6.0 

5- 

^ 

5-4 

4.8 

4.4 

4.0 

3-5 

2.9 

2.2 

i-S 

0.7 

49-7 

34 

36 

4.0 

3- 

S 

3-4 

3-1 

2.7 

2-3 

1.8 

^ 

0.7 

49.9 

49.2 

48-3 

7.2 

36 

38 

2.0 

i. 

S 

•3 

I.O 

0.6 

0.2 

49-7 

49. 

i 

48.4 

7.6 

6.8 

5-8 

4-7 

38 

40 

5O.O 

49-8 

49-3 

48.9 

48.5 

48.0 

47-5 

46.8 

46.1 

45-3 

44-3 

43-3 

42.1 

40 

42 

48.0 

7.8 

7.2 

6.8 

6.4 

5-9 

5-3 

4.6 

3-8 

2.9 

1.9 

0.7 

39-4 

42 

44 

6.0 

5-8 

c 

•  I 

4-7 

4-3 

3-7 

3-1 

2-3 

i 

•5 

0.5 

39-4 

38.1 

6.7 

44 

46 

4.0 

3-8 

>-I 

2.7 

2.1 

0.9 

0.0 

39-i 

38.0 

6.8 

5-4 

3-8 

46 

48 

2.0 

i. 

7 

] 

.O 

0.6 

0.0 

39-3 

38.6 

37.7 

6.7 

5-5 

4.2 

2.7 

0.9 

48  : 

50 
52 

40.0 
38.0 

39- 
7- 

7 
7 

38.9 

6.9 

38.5  |  37-9 
6-3       5-7 

4-9 

36.3 
4.0 

35-4 
3-° 

34.3 
1.8 

33-° 
0.4 

31-5 
28.8 

29.8 
6.8 

27.8 
4-5 

50 
52 

54 

6.0 

5- 

7 

4.8 

4.2 

3-5 

2.7 

i-7 

0.6 

29.2 

27.7 

5-8 

3-6 

0.9 

54 

56 

4.0 

3-7 

2.7 

2.O 

0.4 

29-3 

28. 

i 

6.6 

4.8 

2.7 

O.I 

17.0 

56 

58 

2.0 

1.6 

0.6 

29.9 

29.1 

28.1 

6.9 

5-5 

3-8 

1.8 

19.4 

16.1 

11.7 

58 

60 

3O.O 

29.6 

28.4 

27.7 

26.8 

25-7 

24.4 

22.9 

20.9 

1  8.6 

15.5 

II.2 

o.o 

60 

62 

28.0 

7.6 

6-3 

5-5 

4-5 

3-3 

1.8 

20.0 

17.8 

14.9 

10.8 

0.0 

62 

61 

6.0 

5-6 

4.1 

3-2 

2.1 

0.7 

19.1 

17.0 

14.2 

IO.2 

o.o 

64 

66 

4.0 

3-5 

1.9 

0.9 

197 

18.1 

16.1 

I3.6 

9.9 

0.0 

66 

68 

2.0 

5 

19.7 

1  8.6 

I7.I 

15-3 

12.8 

9.4 

o.o 

68 

70 

20.0 

19.4 

17.4 

16.1 

14.4 

12.1 

8.8 

O.O 

70 

72 

1  8.0 

17.3 

15.0 

i 

3-5 

IJ.4 

8-3 

ao 

72 

74 

16.0 

*5-2 

12.6 

10.7  .|     7.9 

0.0 

74 

76 

14.0 

'3- 

I 

9.8 

7-3  !    0.0 

76 

78 

12.0 

10.9 

6-7 

o.o 

78 

80 

I  O.O 

8. 

7 

o.o 

80 

TABLE  XXVI. 

Limiting  Errors  of  Horizon-Azimuths. 

Dec.  12°. 

Dec.  18°. 

Dec.  24°. 

Dec.  30°. 

Dec.  24°. 

:       Q> 

Partial  Az. 

Partial  Az. 

Partial  Az. 

Partial  Az. 

&      Partial  Az. 

*0 

Error. 

Prob. 

T^it  .1  I 

Error.       Prob. 

rr«.*«i 

Error. 

Prob. 

Tiki    41    1 

Error. 

Prob. 

T  ttiil 

2          Error. 

Prob 

T*\<  11  1 

•tj 

Lat. 

Dec. 

rotai 
Az. 

Lat. 

IUI.ll 

Dec.     Az. 

Lat. 

Dec. 

loiai 
Az. 

Lat. 

Dec. 

1    M  II  I 

Az. 

£       Lat. 

Dec. 

rouu 
Az. 

:    rt 

Error  1 
±12'. 

Error  Error. 

±  W  • 

Error 
±  !2'. 

___ 

Error  Error 

Error 
±12'. 

Error  ] 

irror. 

Error 
±12'. 

Error 

Error 

rt      Error 
n3     ±0°.5 

Error 

Error. 

1           0   . 

0 

0 

0 

0                0 

o 

0 

0 

0 

0 

0 

0              0 

0 

o 

O 

O.O 

to.i 

±0.1 

0.0 

±0.1    ±0.1 

0.0 

±0.1 

±0.1 

0.0 

±0.1 

±0.1 

O       o.o 

±0.5 

±0.5 

30 

0.0 

O.I 

O.I 

±0.1 

O.I         0.2 

±0.1 

O.I 

O.2 

±0.1 

O.I 

0.2 

IO    ±0.1 

0.5 

40 

±0.1 

0.2 

0.2 

O.I 

O.2        O.2 

O.2 

O.2 

0.3 

0.2 

O.I 

0.2 

2O          0.2 

0.5 

o.'6 

50 

O.I 

0.2 

0.2 

0.2 

O.2        0.3 

0.2 

0.2 

0.3 

°-3 

O.2 

0.4 

3O       0.2 

0.6 

0.7! 

60 

O.I 

0.2 

0.2 

0.3 

0.3        0.4 

0-5 

0.4 

0.7 

*o.6 

*o.3 

0.7 

4O       0.3 

0.8 

0.9 

62 

O.2 

o-3 

0.4 

0.4 

0-3        0.5 

0.7 

0.4 

0.8 

ti.o 

to.6 

1.2 

50       0.5 

0.9 

I.O 

64 

O.2 

0.4 

O-5 

0.4        0.7 

I.I 

0.7 

'•3 

60       1.3 

'•7 

2.1 

66 

°'3 

o-3 

0.4 

0.7 

0-5        0.9 

68 

0-3      0.3 

o.  ** 

0.6      1.3 

70 

0.4      0.4 

0.6 

72 

0.5      0.4 

0.7 

74 

76 

0.6  1    0.5 
0.8  ,    0.6 

0.8 

I.O 

*  For  Lat.  55°. 
t  For  Lat.  58°. 

TABLE  XXVIL 


105 


Correction  of  the  observed  Compass-Azimuth  as  taken  on  the 
Apparent  Horizon. 

Declination  of  same  or  contrary  name. 

Lat. 

0° 

5° 

10° 

12° 

14° 

16° 

18° 

20° 

82° 

24° 

26° 

28° 

30° 

Lat. 

o 

0 

0 

o 

o 

0 

0 

0 

o 

0 

0 

0 

0 

0 

o 

O 

0.0 

0.0 

O.O 

o.o 

O.O 

0.0 

0.0 

o.o 

o.o 

0.0 

0.0 

0.0 

0.0 

0 

5 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

5 

10 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

10 

15 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

2 

15 

20 

2 

2 

2 

2 

2 

2 

3 

3 

3 

3 

3 

3 

3 

20 

24 

28 

O. 

3 
3 

0-3 

0-3 

0-3 

0-3 

0.3 

0-3 

o-3 

o-3 

0-3 

0-3 

0.4 
4 

0.4 

4 

24 

28 

32 
36 

S 

5 

5 

* 

1 

| 

5 

5 

1 

I 

§ 

32 
36 

38 

5 

5 

5 

5 

6 

6 

6 

6 

6 

6 

6 

7 

7 

38 

40 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.7 

0.7 

0.7 

0.7 

4O 

42 

6 

6 

6 

6 

6 

7 

7 

7 

7 

8 

8 

8 

42 

44 

6 

6 

7 

7 

7 

7 

7 

7 

8 

8 

8 

9 

9 

44 

46 

7 

7 

7 

7 

7 

8 

8 

8 

8 

9 

9 

9 

I.O 

46 

48 

7 

8 

8 

8 

8 

8 

8 

9 

9 

I.O 

I.O 

I.O 

i 

48 

50 

0.8 

0.8 

0.8 

0.8 

0.9 

0.9 

0.9 

0.9 

I.O 

i.i 

i.i 

i.i 

1.3 

50 

52 

S 

9 

9 

9 

9 

I.O 

I.O 

I.O 

i 

2 

2 

3 

5 

52 

54 

9 

9 

I.O 

I.O 

I.O 

i 

i 

i 

2 

3 

4 

8 

54 

56 

I.O 

I.O 

i 

i 

i 

2 

2 

2 

3 

6 

8 

2.2 

56 

58 

i 

i 

2 

2 

2 

3 

3 

4 

5 

7 

9 

2-3 

3-2 

58 

60 

i. 

2 

1.2 

1.3 

1.3 

1.3 

1.4 

i-S 

1.6 

1.7 

2.0 

2.4 

3-4 

00 

6O 

62 

S 

3 

4 

4 

4 

6 

7 

8 

2.1 

ij 

3-5 

00 

62 

64 

4 

4 

5 

S 

6 

8 

9 

2.2 

6 

3-7 

00 

64 

66 

5 

5 

7 

7 

9 

2.0 

2-3 

S 

3-8 

00 

66 

68 

b 

7 

9 

2.0 

2.2 

4 

9 

4.0 

00 

68 

70 

1.8 

1.9 

2.1 

2-3 

2.6 

3-1 

4-3 

oo 

70 

72 

2.0 

2.1 

5 

8 

'3-3 

4.6 

00 

72 

74 

2. 

5 

3-5 

4.8 

oo 

74 

76 

6        3.0 

8 

00 

76 

78 

3- 

I 

6 

5-7 

00 

78 

80 

3-8 

44 

CO 

8O 

TABLE  XXVII  L                       TABLE  XXIX. 

Error 

of  the  Horizon-  Azimuth  for  an  Error  of 
12',  or  O°.2,  in  the  Latitude. 

Error  of  the  Horizon- 
Azimuth  for  an  Error 

Azimuth. 

of  6',  or  0°.l,  in  the 

90° 

80° 

70° 

60 

50° 

40° 

30° 

20° 

15 

10° 

5° 

0° 

Lat. 

Declination. 

o 

o 

10 

0 

0.0 
0 

0 

0.0 
0 

0 

0.0 
0 

0 

0.0 
0 

0 

0.0 
O 

0 

0.0 
0 

o 
0.0 

I 

o 
O.O 

I 

0 

0.0 

I 

0 

O.O 
2 

0 

0.0 

0 

0.0 

oo 

o 

0 
10 

Position- 
Angle. 

Azimuth. 
Error. 

20 

o 

o 

o 

o 

I 

I 

I 

2 

•7 

4 

8 

oo 

15 

30 

o 

0 

0 

I 

I 

2 

3 

4 

6 

M 

oo 

20 

o 

9O 

o 

o 

35 

0 

o 

I 

I 

2 

2 

4 

5 

8 

6 

00 

25 

8O 

40 

0.0 

0.0 

0. 

0. 

O.I 

0.2 

0-3 

0.5 

0.6 

I.O 

1.9 

00 

40 

7O 

45 

0 

o 

2 

2 

3 

6 

8 

i 

2-3 

00 

45 

60 

5O 

o 

o 

2 

3 

4 

7 

9 

3 

8 

00 

50 

50 

55 

0 

2 

2 

3 

r 

8 

i.i 

6 

3-3 

oo 

55 

60 

o 

2 

3 

4 

6 

9 

3 

2.0 

4.0 

00 

60 

4O 

0« 

O.I 

62 

0.0 

0. 

0. 

0.2 

0-3 

0-5 

0.6 

I.O 

1.4 

2.1 

4-3 

00 

62 

3O 

2 

64 

o 

2 

3 

5 

7 

i 

5 

3 

7 

00 

64 

25 

2 

66 

o 

2 

3 

4 

5 

8 

2 

7 

00 

66 

2O 

68 

0 

2 

3 

4 

6 

9 

4 

8 

8 

7 

oo 

68 

7O 

o 

2 

3 

5 

7 

I.O 

5 

2.0 

3.1 

6.3 

00 

70 

18 

0.3 

72 

0.0 

o. 

0.2 

o.S 

0.8 

i.i 

2-3 

3.5 

7.0 

00 

72 

16 

4 

74 

76 

78 

0 

o 

0 

2 

2 
3 

3 

4 

5 

9 

I.O 

i 

2 

4 
6 

9 

2.2 

6 

6 

9 
4-5 
5-3   ' 

8.0 
9.2 
0.7 

00 

oo 

00 

74 
76 

78 

14 
12 
1O 

4 

I 

8O 

0 

2 

4 

b 

9 

3 

2.0 

3-1 

4.2 

6.4  i 

3-o 

00 

8O 

8 

0.7 

Lat. 

o 

a 

I 

O 

o 

i 

S 

ee 

1 

o 

i 

o 

I 

o 

o 

1 

Lat. 

6 
4 
2 

I.O 

4 
2.9 

Azimuth. 

0 

oo 

106 


TABLE  XXX. 


Time-Azimuths  :  Log  A. 

Latitude. 

Dec. 

Polar 
Dist. 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8> 

9° 

10° 

35 

9-499 

9.488 

9.476 

9.465 

9453 

9.440 

9-427 

9413 

9-399 

9-385 

9-370 

55 

34 

485 

474 

462 

45° 

437 

424 

411 

397 

382 

367 

56 

33 

472 

460 

447 

435 

422 

408 

394 

379 

364 

348 

33  * 

57 

32 

458 

445 

432 

419 

406 

391 

376 

345 

328 

310 

58 

31 

443 

43° 

417 

403 

389 

374 

358 

342 

325 

308 

289 

59 

3O 

9.428 

9415 

9.401 

9-386 

9-371 

9.356 

9-340 

9-323 

9-305 

9.286 

9.267 

60 

29 

413 

399 

384 

369 

354 

337 

320 

302 

284 

264 

243 

61 

28 
27 

397 
380 

382 
365 

367 
349 

333 

335 
316 

3i8 

298 

300 
279 

281 
259 

262 
238 

241 
216 

219 
193 

62 
63 

26 

363 

347 

331 

3H 

296 

277 

257 

236 

214 

190 

165 

64 

25 

9.346 

9.329 

9.312 

9.294 

9-275 

9-255 

9-234 

9.211 

9.188 

9.163 

9.136 

65 

24' 

328 

310 

292 

273 

253 

232 

209 

1  86 

161 

134 

105 

66 

23 

3°9 

290 

271 

251 

230 

207 

184 

159 

132 

103 

072 

67 

22 

289 

269 

249 

228 

206 

182 

'57 

130 

IOI 

070 

036 

68 

21 

268 

248 

226 

204 

1  80 

155 

128 

099 

068 

034 

8.998 

69 

20 

9.246 

9.225 

9.202 

9.179 

9.153 

9.126 

9.097 

9.066 

9.032 

8.996 

8-955 

7O 

19 

224 

20  1 

177 

152 

125 

096 

064 

030 

8-994 

953 

909 

71 

18 

200 

176 

150 

123 

094 

063 

029 

8.992 

952 

907 

857 

72 

17 

175 

149 

122 

093 

06  1 

027 

8.99o 

95° 

905 

855 

798 

73 

16 

I48 

121 

091 

060 

026 

8.989 

948 

903 

853 

796 

730 

74  ! 

15 

9.II9 

9.090 

9-059 

9.025 

8.988 

8.947 

8.902 

8.852 

8-795 

8.728 

8.650 

75 

14 

089 

058 

8.986 

946 

901 

850 

793 

727 

649 

552 

76 

13 

057 

022 

8.985 

944 

899 

849 

792 

725 

647 

551 

427 

77 

12 

022 

8.984 

943 

898 

848 

791 

724 

646 

549 

425 

250 

78 

11 

8.984 

943 

897 

847 

789 

723 

645 

548 

424 

249 

7.948 

79 

10 

8.942 

8.897 

8.846 

8.789 

8.722 

8.643 

8-547 

8.423 

8.247 

7-947 

—  oo 

80 

9 

896 

845 

788 

721 

643 

546 

422 

246 

7.946 

—  oo 

7-947 

81 

8 

845 

787 

721 

642 

545 

421 

245 

7-945 

00 

7.946 

8.247 

82 

7 

787 

720 

641 

545 

420 

244 

7-944 

00 

7-945 

8.246 

423 

83 

6 

719 

641 

544 

419 

244 

7-943 

00 

7-944 

8.245 

422 

547 

84 

5 

8.640 

8-543 

8.419 

8.243 

7.942 

00 

7-943 

8.244 

8.421 

8.546 

8.643 

85 

4 

543 

418 

243 

7-941 

00 

7.942 

8.244 

420 

545 

643 

722 

86  \ 

3 

418 

242 

7.941 

—  oc 

7.941 

8.243 

419 

545 

642 

721 

789 

87  ! 

2 

242 

7.941 

00 

7.941 

8.242 

419 

544 

641 

721 

788 

846 

88 

1 

7.941 

00 

7.941 

8.242 

418 

543 

641 

720 

787 

845 

897 

89 

0 

—  oc 

7.941 

8.242 

8.418 

8-543 

8.640 

8.719 

8.787 

8.845 

8.896 

8.942 

90 

—    1 

7-941 

8.242 

418 

543 

640 

719 

786 

844 

895 

941 

983 

91 

—   2 

8.242 

418 

543 

640 

7^9 

786 

844 

895 

982 

9.020 

92 

—   3 

418 

543 

640 

719 

786 

844 

895 

941 

982 

9.020 

055 

93 

—  4 

543 

640 

719 

786 

844 

895 

940 

982 

9.019 

054 

087 

94 

—   5 

8.640 

8.719 

8.786 

8.844 

8.895 

8.940 

8.982 

9.019 

9-054 

9.086     9.116 

95  1 

—   6 

719 

786 

844 

895 

940 

982 

9.019 

054 

086 

116  1      144 

96 

—   7 

787 

844 

895 

941 

982 

9.019 

054 

086 

116 

144  1      170 

97 

—   8 

895 

941 

982 

9.019 

054 

086 

116 

144 

170  |      194 

98 

—  9 

896 

941 

982 

9.020 

054 

086 

116 

144 

170 

194  |      218 

99 

—10 
—  11 

8.942 
984 

8.983 
9.021 

9.020 

9-°55 
087 

9.087 
117 

9.116 
144 

9.144 
170 

9.170 
194 

9.194 
218 

9.218 
240 

9.240 
261 

100 
101 

—12 

9.022 

056 

088 

117 

H5 

170 

218 

240 

261 

281 

1O2 

-13 

°57 

088 

118 

145 

171 

'95 

218 

240 

201 

281 

300 

103 

—14 

089 

119 

146 

172 

196 

219 

241 

261 

28l 

300 

104 

-15 

9.119 

9.147 

9-173 

9.197 

9.220 

9.241 

9.262 

9.282 

9.301 

9.318 

9-336 

105 

—16 

148 

173 

197 

220 

242 

263 

282 

301 

336 

353 

106 

—17 

198 

221 

243 

263 

283 

302 

320 

337 

353 

369 

107 

—18 

200 

222 

244 

264 

284 

3°3 

320 

337 

354 

385 

1O$ 

—19 

224 

245 

265 

285 

303 

321 

338 

354 

370 

3§ 

400 

1O9 

—20 

9.246 

9.267 

9.286 

9-305 

9.322 

9-339 

9-355 

9-371 

9.386 

9.401 

9415 

110 

—21 

268 

287 

306 

323 

34° 

356 

372 

387 

401 

416 

429 

111 

—22 

289 

307 

325 

357 

373 

388 

402 

416 

430 

443 

112 

—23 
-24 

3°9 

328 

326 

344 

343 
360 

359 
375 

374 
390 

389 
405 

403 
418 

417 

432 

43  i 
444 

444 
457 

469 

113 
114 

90° 

89° 

88° 

87° 

86° 

85° 

84° 

83° 

82° 

81° 

80° 

Dec. 

Polar 
Dist. 

tfo-Latitude. 

TABLE  XXX. 


107 


Time-  Azimuths:  Log  B. 

Latitude. 

Dec. 

Polar 
Dist. 

0° 

1° 

2° 

3° 

4° 

5° 

6° 

7° 

8° 

9° 

10' 

35 

0.501 

0.491 

0.480 

0.470 

0.460 

0.451 

0.442 

o-433 

0.424 

0.415 

0.407 

o 

55 

34 

5H 

5°4 

493 

482 

472 

462 

453 

444 

434 

425 

56 

33 

528 

506 

495 

484 

474 

464 

455 

445 

436 

427 

57 

32 

542 

531 

5*9 

508 

497 

486 

476 

466 

456 

447 

437 

58 

31 

557 

545 

.533 

499 

488 

478 

467 

458 

448 

59 

30 

0.572 

0-559 

0-547 

0-534 

0-523 

0.512 

0.500 

0.490 

0.479 

0.469 

0459 

60 

29 

587 

574 

561 

548 

536 

524 

513 

502 

491 

481 

470 

61 

28 

603 

589 

576 

563 

55° 

"2 

526 

5*5 

503 

493 

482 

62 

27 

620 

605 

578 

564 

539 

528 

516 

5°5 

494 

63 

26 

637 

621 

607 

593 

579 

566 

553 

529 

517 

506 

64 

25 
24 

0.654 
672 

0.638 
656 

0.623 
640 

0.608 
624 

0-594 
610 

0.580 

595 

0.567 
582 

°168 

0.542 
556 

o-530 
543 

0.518 
531 

65  i 
68 

23 

691 

674 

657 

641 

626 

611 

597 

583 

570 

556 

544 

67 

22 

711 

693 

675 

659 

643 

627 

612 

598 

584 

570 

557 

68 

21 

732 

7i3 

694 

677 

660 

644 

628 

613 

599 

585 

69 

20 

0-754 

0-733 

0.714 

0.695 

0.678 

0.661 

0.645 

0.629 

0.614 

0.600 

0.585 

70 

19 

776 

755 

735 

715 

697 

679 

662 

646 

630 

615 

600 

71 

18 

800 

778 

756 

736 

716 

697 

680 

663 

646 

631 

615 

72 

17 

825 

802 

779 

757 

737 

698 

680 

663 

647 

631 

73 

16 

852 

827 

803 

758 

737 

718 

699 

68  1 

664 

647 

74 

15 

0.881 

0-853 

0.827 

0.803 

0.780 

o-759 

0.738 

0.718 

0.699 

0.681 

0.664 

75 

14 

911 

88  1 

854 

828 

804 

759 

739 

719 

700 

682 

76 

13 

943 

912 

882 

855 

829 

805 

782 

760 

739 

719 

700 

77 

12 

978 

944 

912 

883 

855 

829 

805 

782 

760 

739 

719 

78 

11 

1.016 

979 

945 

883 

856 

830 

805 

782 

760 

739 

79 

1O 

1.058 

1.  117 

0.979 

0-945 

0.913 

0.884 

0.856 

0.830 

0.806 

0.782 

0.760 

80 

9 

104 

059 

1.018 

980 

946 

914 

884 

856 

830 

806 

782 

81 

• 

155 

105 

059 

1.018 

981 

946 

914 

884 

856 

830 

806 

82 

7 

213 

156 

105 

059 

1.018 

981 

946 

914 

884 

856 

830 

83 

6 

281 

214 

156 

105 

060 

1.018 

981 

946 

914 

884 

856 

84 

5 

1,360 

1.281 

1.214 

1.156 

1.105 

i.  060 

1.018 

0.981 

0.946 

0.914 

0.884 

85 

4 

457 

360 

281 

214 

156 

105 

060 

1.018 

981 

946 

913 

86 

3 

582- 

360 

281 

214 

156 

I05 

059 

1.018 

980 

945 

87 

2 

758 

582 

457 

360 

281 

214 

156 

105 

059 

1.018 

980 

88 

1 

2.059 

758 

582 

457 

360 

281 

214 

156 

105 

059 

1.017 

89 

0 

+    00 

2.059 

1-758 

1.582 

1-457 

1.360 

1.281 

1.213 

I-I55 

1.104 

1.058 

9O 

•• 

2.059 

+    00 

2.059 

758 

582 

457 

.   359 

280 

213 

155 

103 

91 

—   2 

1.758 

2.059 

+    0° 

2.059 

757 

581 

456 

359 

279 

212 

154 

92 

—   3 

582    1.758 

2.059 

-j-    00 

2.058 

757 

455 

358 

279 

211 

93 

—   4 

457        582 

J-757 

2.058 

+    00 

2.058 

756 

580 

455 

357 

278 

94 

-    5 

1.360     1.457 

1.58! 

1-757 

2.058 

-f    CO 

2.057 

1.756 

1-579 

1-454 

1-357 

95 

—  6 

281        359 

456 

581      1.756 

2.057 

+    00 

2.056 

755 

578 

453 

96 

—    7 

213        280 

359 

455 

580 

2.056 

+    00 

2.055 

754 

577 

97 

—   8 

155 

213 

279 

358 

455 

579 

1-755 

2.055 

+    00 

2.054 

753 

98 

—   9 

104 

155 

212 

279 

357 

454 

578 

J-754 

2.054 

-j-oo 

2-053 

99 

—10 

1.058     1.103 

I.I54 

I.2I1 

1.278 

1-357 

1-453 

1-577 

1-753 

2-053 

+   00 

100 

—11 

016        057 

103 

153 

211 

277 

355 

452 

576 

I-751 

2.052 

1O1 

—12 

0.978        016 

057 

I  O2 

*52 

209 

276 

354 

575 

i.75o 

1O2 

-13 

943     0.978 

015 

056 

IO1 

208 

275        353 

449 

573 

103 

—14 

911        942 

0.977 

OI4 

054 

099 

150 

207 

273 

35  i 

448 

104 

—15 

0.881     0.910 

0.941 

0-975 

I.OI2 

I-°53 

1.098 

1.148 

1.205 

1.272 

i-35o 

105 

-16 

852        879 

909 

940 

0.974 

on 

052 

097 

147 

204 

270 

1O6 

—17 

825    .    851 

878 

907 

939 

o.973 

010 

050 

095 

145 

202 

107 

-18 

800        824 

850 

877 

906 

937 

0.971 

008 

048 

°93 

J43 

108 

—19 

776        799 

823 

848 

875 

904 

936 

0.970 

006 

047 

091 

1O9 

!  -20 

0.754     0.775 

0.798 

0.821 

0.847 

0.874 

0.903 

0-934 

0.968 

1.004 

1.045 

110 

—21 

732        752 

774 

796 

820 

845 

872 

901 

Q32 

0.966 

002 

111 

—22 

711        731 

772 

794 

818 

843 

870 

899 

930 

0.964 

112 

—23 

691        710 

729 

749 

770 

793 

816 

841 

868 

897 

928 

113 

—24 

672 

690 

708 

727 

747 

768 

791 

814 

839 

866 

895 

114 

90° 

89° 

88° 

87°      86° 

85° 

84° 

83° 

82° 

81° 

80° 

Polar 

Co-Latitude. 

Dist. 

108 


TABLE  XXX. 


Time-Azimuths:   Log  A. 

i 

Latitude. 

Dec. 

Polar 
Dist. 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

2O° 

o 

35 

9.370 

9-354 

9-337 

9-320 

9-302 

9.282 

9.262 

9.241 

9.218 

9.194 

9.168 

55 

34 

351 

334 

298 

279 

259 

237 

214 

190 

164 

136 

56  i 

i       33 

331 

295 

276 

255 

234 

211 

1  86 

1  60 

132 

102 

57 

32 

310 

292 

273 

252 

230 

207 

183 

157 

129 

098 

065 

58 

31 

289 

270 

249 

227 

204 

1  80 

153 

125 

:  095 

062 

026 

59 

3O 

9.267 

9.246 

9.224     9.201 

9.176 

9.150 

9-122 

9.092 

9.058 

9.023 

8.983 

60 

29 

243           221 

198 

173 

147 

119 

088 

055 

019 

8.980 

936 

61 

28 

219           195 

171 

144 

116 

085 

052 

016 

8.976 

932 

883 

62 

27 

193 

1  68 

141 

"3 

082 

049 

013 

8-973 

929 

880 

823 

63 

2S 

165  ;     139 

no       080 

046 

OIO 

8.970 

926 

876 

820 

755 

64 

25 

9.136     9.108 

9.077     9.043 

9.007 

8.967 

8.923 

8-873 

8.817 

8.752 

8.674 

65 

24 

105      075 

041 

005 

8.965 

920 

87I 

814 

749 

671 

576 

66  i 

23 

072  ;     039 

002       8.962 

918 

868 

Nil 

746 

668 

573 

449 

67 

22 

036        ooo 

8.960           915 

805 

809 

743 

665  i     570 

446 

8.272 

68 

21 

8.998     8.957 

913           863 

806 

741 

663 

567 

444 

269 

7-969 

69 

20 

8.955  i  8.911 

8.861       8.804 

8.738 

8.660 

8.565 

8.441 

8.266 

7.967 

00 

70 

19 

909        859 

802           736 

658 

562 

439 

264 

7-964 

—  oo 

7.966 

71  ! 

17 

857  i     800 
798  !     732 

734 
654 

656 

558 

560 
434 

436 
259 

7-959 

7.961 

00 

—  oc 
7.961 

7.964 
8.264 

8.266 
441 

72 
73  ! 

IS 

73°       652 

556 

432 

257 

7-957 

—  oo 

7.959 

8.261 

439 

565 

74 

15 

8.650    8.554 

8.430 

8.255 

7-955 

00 

7.957 

8.259 

8.436 

8.562 

8.660 

75 

14 

552       428 

253 

7-953 

00 

7-955 

8.257 

434 

560 

658 

738 

76 

13 

427 

252 

7-95  i 

00 

7-953 

8-255 

432 

558 

736 

804 

77 

12 

250 

7.950 

—  oo 

7-95  i 

8-253 

43° 

556 

654 

734 

802 

86  1 

78 

11 

7.948 

—    00 

7-95° 

8.252 

428 

554 

652 

732 

800 

859 

911 

79 

1O 

00 

7.948 

8.250 

8.427 

8-552 

8.650 

8.730 

8.798 

8.857 

8.909 

8-955 

80 

9 

7-947 

8.249 

425 

551 

649 

728 

796 

855 

907 

953 

996 

81 

8 

8.247 

424 

549 

647 

727 

795 

853 

905 

952 

994 

9.032 

82  i 

7 
6 

423 
547 

548 
645 

646 

724 

725 
792 

793 
850 

852 
902 

903 

948 

950 
990 

992 
9.029 

9.030 
064 

066 
097 

83 
84 

~» 

8.643 

8.723 

8.791 

8.849 

8.901 

8.947 

8.989 

9.027 

9.063 

9.096 

9.126 

85 

4 

722 

-    789 

848 

899 

946 

988 

9.026 

061 

094 

I25 

'53 

86 

3 

789 

847 

898 

944 

986 

9.025 

060 

093 

123 

179 

87 

2 

846 

897 

943 

985 

9.023 

059 

092 

122 

150 

177 

202 

88 

1 

897 

943 

984 

9.022 

058 

090 

121 

149 

176 

2OI 

225 

89 

0 

8.942 

8.984 

9.022 

9-057 

9.089 

9.119 

9.148 

9-175 

9.200 

9.224 

9.246 

90 

—    1 

983 

9.021 

056 

088 

119 

147 

173 

199 

222 

245 

267 

91  i 

—   2 

9.020 

055 

088 

118 

146 

173 

197 

221 

244  !       265 

286 

92 

-   3 
—  4 

oil 

087 
117 

117 

171 

'72 
196 

197 

220 

22O 
242 

243 
263 

264 
284 

285 
303 

305 

322 

93  j 
94 

—   5 

9.116 

9.144 

9.171 

9-195 

9.219 

9.241 

9.263 

9.283 

9-303 

9.321 

9-339 

95 

—  6 

144 

170 

195 

218 

241 

262 

282 

302 

320 

338 

355 

96 

-    * 

170 

195 

218 

240 

261 

282 

3OI 

320 

337 

354 

97 

—   8 

194 

218 

240 

261 

281 

3OI 

319 

337 

354 

370 

386 

98 

-  9 

218 

240 

261 

281 

300 

319 

336 

353 

370 

385 

401 

99 

—10 

9.240 

9.261 

9.281 

9.300 

9.318 

9.336 

9-353 

9-369 

9-385 

9.400 

9-4I5 

100 

—11 

261 

281 

3°° 

3*8 

335 

352 

369 

384 

400 

414 

428 

101 

-12 

281 

300 

3*8 

335 

352 

368 

384 

399 

414 

428 

441 

1O2 

—13 

300 

3i8 

335 

352 

368 

384 

399 

413 

428 

441 

454 

103 

—14 

3*8 

335 

352 

368 

384 

399 

413 

427 

441 

454 

466 

104  | 

-15 

9-336 

9.352 

9.368 

9-384 

9-399 

9-4I3 

9.427 

9.440 

9-454 

9.466 

9.478 

1O5 

—16 

353 

369 

384 

399 

413 

427 

440 

453 

466 

478 

490 

106 

—  17 

369 

384 

399 

413 

427 

440 

453 

466 

478 

490 

502 

107 

—18 

385 

399 

4*4 

427 

441 

453 

466 

478 

490 

502 

108 

—19 

400 

414 

428 

441 

454 

466 

478 

490 

502 

5*3 

524 

1O9 

—20 

9415 

9.428 

9.442 

9-454 

9.466 

9-479 

9.490 

9-502 

9.5I3 

9-524 

9-534 

110 

—21 

429 

442 

455 

467 

479 

491 

502 

513 

524 

534 

544 

111 

—22 
—23 
—24 

443 

£ 

$ 

481 

468 
480 
492 

479 
492 
504 

491 
503 

502 
525 

524 
535 

524 
535 
545 

535 
545 
555 

544 
554 
564 

564 
574 

112  j 
113 
114 

80° 

79° 

78° 

77D 

76° 

75° 

74° 

73° 

72° 

71° 

70° 

i 

—  » 

Polar 

Co-Latitude. 

Dist. 

TABLE  XXX. 


109 


Time-Azimuths:  Log  15. 

Latitude. 

TX^  _ 

Polar 

Dec. 

10° 

11° 

12° 

13° 

14° 

15° 

16° 

17° 

18° 

19° 

20° 

Dist. 

o 

35 

0.407 

0.398 

0.390 

0-383 

0-375 

0.367 

0.360 

o-353 

0.346 

0-339 

0.332 

0 

55 

34 

4*7 

408 

400 

392 

384 

376 

369 

361 

354 

347 

340 

56 

33 

427 

418 

410 

401 

393 

385 

378 

370 

362 

355 

348 

57 

32 

437 

429 

420 

411 

403 

394 

387 

363 

356 

58 

31 

448 

439 

43° 

421 

412 

4°4 

396 

388 

379 

372 

364 

59 

3O 

0-459 

0.450 

0.440 

0.431 

0.422 

0.413 

0.405 

0-397 

0.388 

0.380 

0.372 

60 

29 

470 

461 

45  * 

442 

432 

423 

414 

406 

397 

389 

381 

61 

28 

482 

472 

462 

452 

442 

433 

424 

415 

406 

398 

390 

62 

27 

494 

483 

473 

463 

453 

443 

434 

425 

416 

407 

399 

63 

26 

506 

495 

484 

474 

463 

454 

444 

435 

425 

416 

408 

64 

25 

0.518 

0.507 

0.496 

0.485 

0-474 

0.464 

0-454 

0-445 

0435 

0.426 

0.417 

65 

24 

531 

5J9 

508 

496 

486 

475 

465 

455 

445 

436 

426 

66 

23 

544 

520 

508 

497 

486 

476 

465 

455 

446 

436 

67 

22 

557 

545 

532 

520 

5°9 

498 

487 

476 

466 

456 

446 

68 

21 

558 

545 

533 

521 

510 

498 

487 

476 

466 

456 

69 

20 

0.585 

0.572 

0-559 

0.546 

0.533 

0.522 

0.510 

0.498 

0.487 

0.476 

0.466 

70 

19 

600 

586 

572 

559 

546 

534 

522 

510 

498 

487 

476 

71 

18 

615 

60  1 

586 

573 

559 

547 

534 

522 

5IQ 

498 

487 

72 

17 

616 

60  1 

573 

560 

547 

534 

522 

51° 

498 

73  i 

16 

647 

631 

616 

601 

587 

573 

560 

547 

534 

522 

74 

15 

0.664 

0.648 

0.632 

0.616 

0.60  1 

0.587 

0-573 

0.560 

0.546 

0-534 

0.522 

75 

14 

682 

665 

648 

632 

616 

60  1 

587 

573 

559 

546 

534 

76 

13 

700 

682 

665 

648 

632 

616 

601 

573 

559 

546 

77 

12 

719 

700 

682 

665 

648 

632 

616 

60  1 

587 

559 

78 

11 

739 

719 

700 

682 

665 

648 

631 

616 

601 

586 

572 

79 

1O 

0.760 

0-739 

0.719 

0.700 

0.682 

0.664 

0.647 

0.631 

0.615 

0.600 

0.585 

80 

9 

782 

760 

739 

719 

700 

681 

664 

647 

630 

615 

599 

81 

8 

806 

782 

760 

739 

719 

699 

68  1 

663 

646 

630 

614 

82 

7 

830 

805 

782 

760 

739 

718 

699 

680 

663 

646 

629 

83 

6 

856 

830 

805 

782 

759 

738 

718 

698 

680 

662 

645 

84 

5 

0.884 

0.856 

0.829 

0.805 

0.781 

0.759 

0-737 

0.717 

0.698 

0.679 

0.661 

85 

4 

3 

913 

883 
913 

855 
883 

829 
855 

804 
828 

I 

g 

737 

757 

716 

736 

697 
715 

678 
695 

86 
87 

2 

980 

945 

912 

882 

854 

827 

803 

779 

756 

735 

88  ! 

1 

1.017 

979 

944 

912 

882 

853 

827 

802 

778 

755 

733 

89 

O 

1.058 

1.016 

0.978 

0-943 

0.911 

0.881 

0.852 

0.826 

0.800 

0.776 

0-754 

90 

—    1 

103 

057 

1.016 

978 

942 

910 

879 

851 

824 

799 

775 

91 

—   2 

154 

103 

056 

1.015 

977 

94i 

909 

878 

850 

823 

798 

92 

—   3 

211 

153 

102 

056 

1.014 

975 

940 

907 

877 

848 

821 

93 

—  4 

278 

211 

152 

101 

054 

I.OI2 

974 

939 

906 

875 

847 

94 

—   5 

1-357 

1.277 

1.209 

1.151 

1.099 

1-053 

I.OII 

0-973 

0-937 

0.904 

0.874 

95 

-   6 

453 

355 

276 

208 

150 

098 

052 

I.OIO 

971 

936 

903 

96 

-    7 

577 

452 

354 

275 

207 

148 

097 

050 

1.008 

970 

934 

97 

—   8 

753 

576 

353 

273 

205 

095 

048 

i.  006 

968 

98 

—   9 

2-053 

575 

449 

35  i 

272 

204 

145 

093 

047 

1.004 

99 

—10 

+  oc 

2.052 

i-75° 

1-573 

1.448 

1-350 

1.270 

1.202 

I-I43 

1.091 

1.045 

1OO 

—11 

2.052 

+    00 

2.050 

748 

572 

446 

348 

268 

200 

141 

089 

101 

—12 

i-75° 

2.050 

-f-    00 

2.049 

747 

570 

444 

346 

266 

198 

139 

102 

—13 

573 

1.748 

2.049 

+  oo 

2.047 

745 

568 

442 

344 

264 

196 

1O3 

—14 

448 

572 

1-747 

2.047 

+   00 

2.045 

743 

566 

440 

342 

262 

104 

—15 
—16 

270 

1.446 

348 

1-570 
444 

1-745 
568 

2-045 
1-743 

+    00 

2.043 

2.043 

-4-   00 

1.741 
2.041 

1.564 
739 

1.438 
561 

1.340 
435 

1O5 

106 

—17 

202 

268 

346 

442 

566 

1.741 

2.041 

-f  oo 

2.039 

736 

559 

107 

—18 

H3 

200 

266 

344 

440 

564 

1-739 

2.039 

-f-  oo 

2.036 

734 

108 

—19 

091 

141 

198 

264 

342 

438 

561 

i-736 

2.036 

-f    00 

2.033 

1O9 

-20 

1.045 

1.089 

I-I39 

1.196 

1.262 

1.340 

1-435 

1-559 

1-734 

2.033 

4-  oo 

110 

—21 

002 

043 

087 

137 

194 

259 

337 

433 

556 

1-731 

2.031 

111 

—22 

0.964 

ooo 

040 

085 

J35 

191 

257 

335 

43° 

554 

1.728 

112 

-23 

928 

0.961 

0.998 

038 

082 

132 

189 

254 

332 

427 

551 

113 

—24 

895 

925 

959 

0-995 

035 

080 

129 

186 

251 

329 

424 

114 

Dec 

80° 

79° 

78° 

77° 

76° 

75° 

74° 

73° 

72° 

71° 

7O° 

Polar 

Diat 

Co-Latitude. 

XJlOl* 

110 


TABLE  XXX. 


Time-Azimuths:  JLogr  A. 

Latitude. 

Dec. 

Polar 
Dist. 

20° 

21° 

22° 

23° 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

35 

9.168 

9.140 

9.110 

9.077 

9.042 

9.003 

8-959 

8.911 

8.855 

8.790 

8.714 

55 

34 

136 

106 

073 

038 

8.998 

8-955 

906 

850 

786 

709 

614 

56 

33 

1  02 

069 

034 

8-994 

95  i 

902 

846 

781 

704 

610 

487 

57 

32 

065 

030 

8.990 

947 

898 

842 

777        7oo 

60; 

483 

3°9 

58 

31 

026 

8.987 

943 

894 

838 

773 

696 

60  1 

478 

3°4 

006 

59 

30 

8.983 

8-939 

8.890 

8.834 

8.769 

8.692 

8-597 

8-474 

8.300 

8.001 

00 

6O 

29 

936 

886 

830 

765 

688 

593 

470 

296 

7-997 

OO 

8.001 

61 

28 

883 

827 

762 

684 

589 

•466 

292 

7-993 

—  oo 

7-997 

300 

62 

27 

823 

758 

681 

586 

462 

288 

7-989 

00 

7-993 

8.296 

474 

63 

:         26 

755 

677 

582 

459 

285 

7-985 

—  oo 

7.989 

8.292 

470 

597 

61 

25 

8.674 

8-579 

8.456 

8.281 

7.982 

00 

7-985 

8.288 

8.466 

8-593 

8.692 

65 

24 

576 

452 

278 

7-978 

00 

7.982 

8.285 

462 

589 

688 

769 

66 

23 

449 

275 

7-975 

—  oo 

7.978 

8.281 

459 

586 

684 

765 

834 

67 

22 

272 

7-972 

00 

7-975 

8.278 

456 

582 

681 

762 

830 

890 

68 

21 

7.969 

—  oo 

7-972 

8.275 

452 

579 

677 

758 

827 

886 

939 

69 

20 

00 

7.969 

8.272 

8-449 

8.576 

8.674 

8-755 

8.823 

8.883 

8.936 

8.983 

70 

19 

7.966 

8.269 

446 

573 

671 

752 

820 

880 

932 

980 

9.023 

7.1 

18 

8.266 

444 

57° 

668 

749 

817 

876 

929 

976 

9.019 

059 

72 

17 

441 

567 

665 

746 

814 

873 

926 

973 

9.016 

°55 

092 

73 

16 

565 

663 

743 

811 

871 

923 

9/0 

9.013 

052 

088 

122 

74  i 

15 

8.660 

8.741 

9.809 

8.868 

8.920 

8.967 

9.010 

9.049 

9-085 

9.119 

9.150 

75  ! 

14 

738 

806 

865 

918 

965 

9.007 

046 

082 

116 

147 

I76 

76 

13 

804 

863 

915 

962 

9.005 

°44 

080 

113 

144 

173 

201 

77  ! 

12 

861 

913 

960 

9.002 

041 

077 

no 

141 

171 

198 

224 

78  I 

11 

911 

957 

9.000 

039 

075 

108 

139 

1  68 

195 

221 

246 

79  j 

10 

8.955 

8.998 

9.036 

9.072 

9.105 

9.136 

9.165 

9-193 

9.219 

9-243 

9.267 

80  i 

9 

996 

9.034 

070 

103 

134 

163 

190 

216 

241 

264 

286 

81 

8 

9.032 

068 

101 

132 

161 

188 

214 

238 

262 

284 

3°5 

82 

7 

066 

099 

130 

186 

212 

236 

259 

281 

302 

323 

83 

6 

097 

128 

157 

184 

209 

234 

257 

279 

300 

320 

340 

84 

5 

9.126 

9-155 

9.182 

9.207 

9.232 

9-255 

9.277 

9.298 

9.318 

9-337 

9-356 

85 

4 

153 

1  80 

206 

230 

253 

275 

296 

316 

335 

354 

372 

86 

3 

179 

204 

228 

251 

273 

294 

3i4 

333 

351 

369 

387 

87 

2 

202 

226 

249 

271 

292 

3I2 

33i 

349 

367 

384 

401 

88 

1 

225 

248 

269 

290 

310 

329 

348 

365 

382 

399 

415 

89 

0 

9-246 

9.268 

9.289 

9-309 

9-328 

9.346 

9-363 

9.380 

9-397 

9-4I3 

9428 

90 

—    1 

267 

287 

307 

326 

344 

362 

379 

395 

411 

426 

441 

91 

—   2 

286 

306 

325 

343 

360 

377 

393 

409 

424 

439 

453 

92 

—   3 
—  4 

305 
322 

323 
340 

357 

359 
374 

376 

392 
406 

407 
421 

423 
436 

437 
45° 

464 

466 
477 

93 
94 

-   5 

9-339 

9-373 

9-389 

9-405 

9.420 

9-434 

9.448 

9.462 

9.476 

9.489 

95 

—  6 

355 

372 

388 

403 

418 

433 

447 

461 

474 

487 

500 

96 

—   7 

37i 

387 

402 

417 

432 

446 

459 

473 

485 

498 

5IQ 

97 

—   9 

386 

401 

416 

431 

445 

458 

47i 

484 

497 

509 

521 

98 

—  9 

401 

415 

430 

444 

457 

470 

483 

495 

508 

519 

531 

99 

—10 

9-4I5 

9.429 

9-443 

9-456 

9.469 

9.482 

9.494 

9.506 

9.518 

9-530 

9-541 

100 

—11 

428 

442 

455 

468 

481 

493 

505 

5*7 

528 

540 

550 

1O1 

—12 

441 

455 

467 

480 

492 

5°4 

527 

538 

549 

560 

102 

—13 

454 

467 

479 

492 

5°3 

526 

537 

548 

559 

569 

1O3 

—14 

466 

479 

491 

503 

5H 

526 

537 

547 

558 

568 

578 

104 

—15 

9.478 

9.491 

9.502 

9.SH 

9.525 

9.536 

9.546 

9-557 

9.567 

9-577 

9-587 

105 

—16 

490 

502 

513 

524 

535 

546 

556 

566 

576 

586 

595 

1O6 

—17 

502 

513 

524 

535 

545 

555 

565 

575 

585 

594 

604 

1O7 

-18 

513 

524 

534 

545 

555 

565 

575 

584 

594 

603 

612 

1O8 

—19 

524 

534 

544 

555 

564 

574 

584 

593 

602 

611 

620 

10?) 

i  —20 

9-534 

9-544 

9-554 

9-564 

9-574 

9.583 

9-593 

9.602 

9.610 

9.619 

9.628 

110 

—21 

544 

554 

564 

574 

583 

592 

60  1 

610 

619 

627 

635 

111 

—22 

554 

564 

574 

583 

592 

60  1 

610, 

618 

627 

635 

643 

112 

-23 

564 

574 

583 

592 

601 

609 

618 

626 

634 

642 

650 

113 

—24 

574 

583 

592 

601 

609 

618 

626 

634 

642 

650 

658 

114 

Dec. 

70° 

69° 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

61° 

60° 

Polar 

Co-Latitude. 

Dist. 

TABLE  XXX. 


Ill 


Time-Azimuths  :  Log  B. 

Latitude. 

Polar 

Dec 

20° 

21°  i  22°   23° 

24° 

25° 

26° 

27° 

28° 

29° 

30° 

Dist. 

o 

35 

0-332 

|     j 
0.325  0.319  0.312 

'0.306 

0.299 

0.293 

0.287 

0.281 

0-275 

0.269 

55 

34 

34° 

333  '  326 

319 

3*3 

306 

300 

294 

288 

282 

276 

56 

33 

348 

34i   333 

326 

320 

313 

3°7 

300 

294 

288 

282 

57 

32 

356 

349  i  34i   334 

327 

320 

3H 

307 

301 

294 

288 

58 

31 

364 

357   349   342 

335 

328 

321 

3H 

308 

301 

295 

59 

30 

0.372 

0.365  0.357  1  0.350 

0.342 

0-335 

0.328 

0.321 

0.314 

0.308 

0.301 

60 

29 

381 

373 

365 

358 

350 

343 

335 

328 

321 

314 

308 

61 

39° 

373 

366 

358 

350 

343 

336 

328 

321 

3H 

62 

27 

399 

390   382 

374 

366 

358 

350 

343 

336 

328 

321 

63 

26 

408 

399   390 

382 

374 

366 

358 

350 

343 

335 

328 

64 

25 

0.417 

0.408  !  0.399 

0.391 

0.382 

0-374 

0.366 

0.358 

0.350 

0-342 

0-335 

65 

24 

426 

417   408 

399 

391 

382 

374 

366 

358 

350 

342 

66 

1   23 

436 

426  !   417 

408 

399 

39  1 

382 

374 

366 

358 

350 

67 

22 

446 

436    426 

417 

408 

399 

390 

382 

374 

365 

357 

68 

21 

456 

446   436 

426 

417 

408 

399 

390 

382 

373 

365 

69  1 

20 

0.466 

0.456  0.446  0.436 

0.426 

0.417 

0.407 

0-399 

0.390 

0.381 

0.372 

70 

19 

476 

466  j  456  i  445 

435 

426 

416 

407 

398 

389 

380 

71 

18 

487 

476  \  466  i  455   445 

435 

425 

416 

406 

397 

388 

72 

498 

487   476   465 

455 

445 

435 

425 

415 

406 

396 

73 

16 

498  i  487   476 

465 

454 

444 

434 

424 

414 

405 

74 

15 

0.522 

0.509  i  0.498  0.486 

0-475 

0.464 

0-454 

0-443 

0-433 

0.423 

0.413 

75  : 

14 

534   521  :  509  j  497 

486 

474 

464 

453 

442 

432 

422 

76 

13 

546   533  i  520   5°8 

497 

485 

474 

463 

452 

441 

77 

12 

559 

545 

532 

520 

508 

496 

484 

473 

462 

451 

440 

78 

11 

572 

558 

544 

532 

519 

5°7 

495 

483 

472 

461 

450 

79 

10 

0.585 

o.57i 

0-557 

0-544 

0.531 

0.518 

0.506 

0.494 

0.482 

0.471 

0459 

8O 

9 

599 

585 

556 

543 

5^0 

517 

505 

492 

481 

469 

81 

8 

614 

599 

584 

569 

542 

529 

516 

5°3 

491 

479 

82 

7 

629 

613 

598 

583 

568 

554 

541 

527 

515 

502 

490 

83 

6 

645 

628 

612 

597 

581 

567 

553 

539 

526 

513 

500 

84 

5 

0.661 

0.644 

0.627 

o.6n 

0-595 

0.580 

0.566 

0-552 

0.538 

0.524 

0.511 

85 

4 

678 

660 

643 

626 

609 

594 

579 

564 

55° 

536 

523 

86 

3 

695 

677 

659 

641 

624 

608 

593 

577 

548 

535 

-  87 

2 

694   675 

657 

640 

623 

607 

576 

561 

547 

88  ! 

1 

733 

712 

693 

674 

656 

638 

621 

605 

589 

574 

559 

89 

0 

0-754 

0.732 

0.711 

0.691 

0.672 

0.654 

0.637 

0.620 

0.603 

0.587 

0.572 

90 

1  —  1 

775 

752 

73i 

710 

690 

671 

653 

635 

618 

60  1 

585 

91 

-  2 

798 

774 

75  i 

729 

708 

688 

669 

651 

633 

616 

599 

92 

—  3 

821 

796 

772 

749 

727 

706 

686 

667 

649 

631 

614 

93 

—  4 

847 

820 

794 

770 

747 

725 

704 

684 

665 

646 

629 

94 

—  5 

0.874 

0.845 

0.818 

0-793 

0.768 

0-745 

0.723 

0.702 

0.682 

0.663 

0.644 

95 

—  6 

903 

872 

843 

816 

791 

766 

743 

721 

700 

680 

660 

96 

—  7 

934 

901 

870 

841 

814 

789 

764 

719 

698 

677 

97 

—  8 

968 

932 

899 

868 

839 

812 

786 

762 

738 

716 

695 

98 

—  9 

1.004 

966 

93° 

897 

866 

837 

810 

784 

759 

736 

7H 

99 

—  1O 

1.045 

1.002 

0.964 

0.928 

0.895 

0.864 

0-835 

0.807 

0.781 

0-757 

0-733 

100 

—11 

089 

043 

1.  000 

96i 

925 

892 

861 

832 

805 

779 

101 

—12 

139 

087 

040 

998 

959 

923 

890 

859 

829 

802 

776 

102 

-13 

196 

137 

085 

1.038 

995 

957 

920 

887 

856 

827 

799 

103 

—14 

262 

194 

135 

082 

1-035 

993 

954 

918 

884 

853 

824 

1O4 

-15 

1.340 

1.259 

1.191 

1.132 

1.080 

I-°33 

0.990 

0.951 

0.915 

0.881 

0.850 

105 

-16 

435 

337 

257 

189 

129 

077 

1.030 

987 

948 

912 

878 

106 

—17 

559 

433 

335 

254 

1  86 

126 

074 

1.027 

984 

945 

908 

107 

—  1§ 

734 

55i 

430 

332 

25  1 

183 

123 

071 

1.024 

981 

942 

108 

—19 

2.033 

554 

427 

329 

248 

180 

120 

068 

1.020 

977 

109 

-20 

+  00 

2.031 

1.728 

I-55I 

1.424 

1.326 

1.245 

I.I77 

1.117 

1.064 

1.017 

110 

—21 

2.031 

+  00 

2.028 

725 

548 

421 

323 

242 

173 

114 

06  1 

111 

—22 

1.728 

2.028 

+  00 

2.025 

722 

544 

418 

319 

238 

170 

no 

112 

—23 

551 

1-725 

2.025 

+  00 

2.O22 

719 

54i 

414 

316 

235 

166 

113 

—24 

424 

548 

1.722 

2.022 

+  00 

2.018 

538 

411 

3I2 

231 

114 

Dec. 

70° 

69° 

68° 

67° 

66° 

65° 

64° 

63° 

62° 

61° 

60° 

Polar 

Co-Latitude. 

Dist. 

112 


TABLE  XXX. 


Time-Azimuths  :  Log  A. 

Latitude. 

Dec. 

353 

•* 

Polar 
Dist. 

30° 

31 

3i>   33 

34° 

37   38°   39a  40° 

0 

35 

8.714 

8.619 

8.497 

8.3-3  8-025 

CO 

8.030 

8-334  S.SM  8.641  :  8.740 

o 

&S 

34 

614 

492 

3'8 

8.020  —  co 

8.025 

329 

5°7   635    7^   Si  7 

56 

33 

487 

3H 

8.015 

00 

8.O2O 

323 

502 

.629 

729   Si  i  i   S8i 

57 

32 

3°9 

8.010 

00 

8.015 

318 

497 

624 

724 

805 

8?5 

936 

5N 

31 

8.006 

00 

8.010 

3!3 

492 

619 

719 

800 

870 

93° 

984 

59 

30 

—  oo 

8.006 

8.308 

8.487 

8.614 

8.714 

8.795 

8.865 

8.925 

8-979 

9.027 

60 

29 

8.001 

3°4 

483 

610 

709 

790 

860 

920 

974 

9.022 

066 

61 

*§ 

300 

478 

605 

704 

786 

855 

9i5 

969 

9.017 

06  1 

IOI 

«2 

27 

474 

601 

700 

781 

850 

911 

954 

9.012 

056 

096 

133 

63 

26 

597 

696 

777 

846 

906 

959 

9.007 

051 

091 

128 

162 

64 

25 

8.692 

8-773 

8.842 

8.902 

8-955 

9.003 

9.046 

9.086 

9.123 

9.158 

9.190 

65 

24 

769 

838 

898 

95i 

999 

042 

082 

119 

153 

185 

2I5 

66 

23 

834 

894 

947 

994  9-038 

077 

114 

148 

180 

211 

239 

67 

22 

890 

943 

990 

9.034   073 

IIO 

144 

176 

206 

234 

261 

6$ 

21 

939 

987 

9.030 

069 

IOJ 

140 

172 

202 

230 

257 

282 

69 

20 

8.983 

9.026 

9.066 

9.102 

9.136 

9.168 

9.198  9.226 

9-253 

9.278 

9.302 

7O 

19 

9.023 

062 

098 

132 

164 

194 

222     248 

274 

298 

321 

71 

IS 

059 

095 

129 

1  60 

190 

218 

244 

270 

294 

317 

339 

72 

17 

092 

I25 

157 

186 

214 

241 

265 

290 

3J3 

335 

356 

73 

16 

122 

153 

183 

211 

237 

262 

286 

309 

33i 

352 

372 

74 

15 

9.150 

9.180 

9.207 

9-234 

9-259 

9.282 

9-3°5 

9.327 

9-348 

9.368 

9-387 

75  : 

14 

I76 

204 

230 

255 

2  79 

302 

323 

344 

364 

384 

402 

76 

13 

201 

227 

252 

276 

298 

320 

34i 

361 

380 

399 

416 

77 

12 

224 

249 

273 

295 

317 

337 

357 

376 

395 

413 

43° 

78 

11 

246 

270 

292 

3U 

334 

354 

373 

39i 

409 

427 

443 

79 

10 

9.267 

9.289 

9.310 

9.331 

9-35  J 

9-370 

9.388 

9.406  j  9.423 

9.440 

9-456 

80 

9 

286 

3oS 

328 

348 

367 

385 

403 

420 

436 

452 

468 

81 

§ 

3°5 

325 

345 

364 

382 

400 

417 

433 

449 

465 

480 

82 

7 

323 

342 

361 

379 

397 

414 

430 

446 

461 

476 

49  i 

83 

6 

34° 

358 

376 

394 

411 

427 

443 

458 

473 

483 

502 

84 

5 

9-356 

9-374 

9-39  i 

9.408 

9-424 

9.440 

9-455 

9.470 

9485 

9-499 

9-5I3 

85 

4 

372 

389 

406 

422 

437 

453 

467 

482 

496 

5io 

523 

86 

3 

387 

403 

419 

435 

45° 

465 

479 

493 

507 

520 

533 

87 

2 

401 

417 

432 

448 

462 

476 

490 

504 

517 

530 

542 

88 

1 

415 

430 

445 

460 

474 

488 

5oi 

5H 

52/ 

540 

552 

89 

0 

9.428 

9-443 

9-457 

9-472 

9-485 

9-499 

9-512 

9-524 

9-537 

9-549 

9.561 

90 

—  1 

441 

455 

469 

483 

496 

509 

522 

534 

547 

558 

570 

91 

—  2 

453 

467 

481 

494 

507 

520 

532 

544 

556 

567 

579 

92 

—  3 

465 

479 

492 

5°5 

518 

530 

542 

554 

565 

576 

587 

93 

—  4 

477 

490 

5°3 

5i6 

528 

54° 

551 

563 

574 

585 

595 

94 

—  5 

9.488 

9.501 

9-5I4 

9.526 

9.538 

9-549 

9.560 

9-572 

9.582 

9-593 

9.603 

95 

g 

499 

512 

524 

536 

547 

558 

569 

580 

59i 

60  1 

611 

96 

—  7 

510 

522 

534 

545 

557 

567 

578 

589 

599 

609 

619 

97 

—  8 

—  ft 

.521 
531 

532 
542 

544 
553 

564 

566 
575 

576 
585 

587 
595 

605 

607 
615 

6i7 
625 

627 
634 

98 
99 

!  —10 

9-54* 

9-552 

9.562 

9-573 

9-583 

9-593 

9.603 

9.613 

9.623 

9.632 

9.641 

100 

i  —11 

551 

56i 

57i 

582 

592 

602 

611 

621 

630 

639 

648 

1O1 

—12 

560 

570 

580 

590 

600 

610 

619 

628 

637 

646 

655 

1O2 

-13 

569 

579 

589 

599 

608 

617 

627 

636 

644 

653 

662 

10* 

—14 

578 

588 

597 

607 

616 

625 

634 

643 

651 

660 

668 

104 

-15 

9-587 

9-596 

9.606 

9.615 

9-624 

9-633 

9.641 

9.650 

9.658 

9.667 

9-675 

105 

—16 

595 

605 

614  ' 

623 

631 

640 

648 

657 

665 

673 

68  1 

106 

—17 

604 

613 

622 

630 

639 

647 

655 

664 

671 

6So 

688 

107 

—18 

612 

62! 

629 

638 

646 

654 

662 

670 

678 

686 

694 

1O8 

—19 

620 

628 

637 

645 

653 

66  1 

669 

677 

685 

692 

700 

109 

—20 

9.628 

9.636 

9.644 

9.652 

9.660 

9668 

9.676 

9.684 

9.691 

9.698 

9.706 

110 

—21 

635 

644 

652 

659 

667 

675 

683 

690 

697 

704 

712 

111 

—22 

643 

651 

659 

666 

674 

682 

689 

696 

703 

710 

717 

112 

-23 

650 

658 

666 

673 

68  1 

688 

695 

702 

709 

716 

723 

113 

—24 

! 

658 

665 

673 

680 

687 

694 

701 

708 

715 

722 

728 

114 

60° 

59° 

5§° 

57° 

56° 

55° 

54° 

53° 

52° 

51° 

50° 

Polar 

Dsc. 

! 

Co-Latitude. 

Dist. 

TABLE  XXX. 


113 


Time-  Azimuths:  Log  B. 

Latitude. 

Polar 

Dec. 

30°   31° 

32° 

33°   34° 

35° 

36°   37° 

38° 

39° 

40° 

Dist. 

1   35 

0.269  0.264 

0.258 

| 
0.252  i  0.247 

0.241 

0.236  i  0.231 

0.226 

O.22O 

0.215 

55  : 

34 

276   270 

264 

258  !  252 

247 

241   236   231   225 

220 

56 

33 

282   276 

270 

264 

258 

252 

247   241 

236  I   2^0 

225 

57 

32 

288   282 

276 

270 

264 

258 

252    247   241    2^5 

230 

58 

i   31 

295   288 

282 

276 

270 

264 

258   252 

246     240 

235 

59 

30 

0.301  0.295 

0.288 

0.282 

0.276 

0.269 

0.263 

0-257 

0.251   0.246 

0.240 

60 

29 

308   301 

294 

288 

282 

275 

269 

263 

257 

251 

245 

61 

28 

3H   308 

301 

294 

288 

281 

275 

269 

262 

256 

250 

62 

27 

321 

314 

307 

294 

287 

281 

274 

268 

261 

255 

63 

26 

328 

321 

3H 

307 

300 

293 

287 

280 

273 

267 

261 

64 

25 

0.335  0.328 

0.320 

0-313 

0.306 

0.299 

0.293 

0.286 

O.279 

0.273 

0.266 

65 

24 

342   335 

327 

320 

313 

306 

299 

292 

285 

278 

272 

66 

23 

350   342 

334 

327 

319 

312 

305 

298 

291 

284 

67 

22 

357 

349 

341 

334 

326 

318 

3" 

3°4 

297     290 

283 

68 

21 

365 

356 

348  I  34i 

333 

325 

318 

310 

3°3 

296 

289 

69 

23 

0.372 

0.364 

°-356  0.348 

0.340 

0.332 

0.324  0.316 

0.309 

0.302 

0.294 

70 

19 

380 

372 

363 

355 

347 

339 

331 

323 

315 

308 

300 

71 

18 

388 

380 

37i 

362 

354 

346 

338 

330 

322 

314 

306 

72 

17 

396 

388 

379 

37o 

361 

353 

345 

336 

328 

320 

312 

73 

16 

405 

396 

387 

377 

369 

360 

352 

343 

335 

327 

319 

74 

15 

0.413 

0.404 

Q-394 

0-385 

0.376 

0.367 

0-359 

0-350 

0.342 

0.333 

0-325 

75 

14 

422 

412 

403 

393 

384 

375 

366 

357 

349 

34° 

332 

76 

13 

43^ 

421 

411 

401 

392 

f3 

373 

364 

356 

347 

338 

77 

12 

440 

43° 

420 

410 

400 

381   372 

363    354 

345 

78  ; 

11 

45° 

439 

429 

418 

408 

399 

389   379 

370 

361 

352 

79 

10 

0-459 

0.448 

0.438 

0.427 

0.417 

0.407 

0-397 

0.387 

0.378 

0.368 

0-359 

80 

9 

469 

458 

447 

436 

426 

415 

405 

395 

385 

376 

366 

81 

8 

479 

468 

456 

445 

435 

424 

413 

403 

393 

383 

374 

82 

7 

490 

478 

466 

455 

444 

433 

422 

411 

401 

391 

83 

6 

500 

488 

476 

464 

453 

442 

420 

409 

399 

389 

84  ! 

5 

0.511 

0-499 

0.486 

0-474 

0.462 

0-451 

0.440 

0.428 

0.418 

0.407 

0-397 

85  ! 

4 

523 

497 

484 

472 

460 

449 

437 

426 

405 

86 

3 

535 

52i 

508 

495 

482 

470 

458 

447 

435 

424 

413 

87 

2 

547 

533 

519 

566 

493 

480 

468 

456 

444 

433 

421 

88 

1 

559 

545 

531 

5r7 

504 

491 

478 

466 

453 

442 

43° 

89  : 

O 

0.572 

0-557 

0-543 

0.528 

0.515 

0.501 

0.488 

0.476 

0.463 

0.451 

0-439 

90 

—  I 

585 

570 

555 

540 

526 

512 

499 

486 

473 

460 

448 

91 

—  2 

599 

583 

568 

552 

538 

524 

496 

483 

470 

458 

92 

—  3 

614 

597 

581 

565 

55° 

535 

521 

507 

493 

480 

467 

93 

—  4 

629 

611 

594 

578 

563 

547 

533 

518 

5°4 

491 

477 

94 

—  5 

0.644 

0.626 

0.609 

0.592 

0.576 

0.560 

0-545 

0.530 

0.515 

0.501 

0.488 

95 

—  6 

660 

642 

624 

606 

589 

573- 

557 

542 

527 

512 

498 

96 

—  7 

677 

658 

639 

621 

603 

587 

570 

554 

539 

524 

5°9 

97 

—  8 

695 

675 

655 

636 

618 

60  1 

584 

567 

536 

520 

98 

—  9 

7H 

692 

672 

652 

633 

615 

598 

580 

564 

548 

•532 

99 

—  1O 

0-733 

0.711 

0.690 

0.669 

0.649 

0.630 

0.612 

0-594 

0-577 

0.560 

o-544 

100 

—11 

754 

73° 

708 

687 

666 

646 

627 

609 

59i 

'573 

557 

101 

—12 

776 

727 

705 

683 

663 

643 

624 

605 

570 

102 

—13 

799 

773 

748 

724 

702 

680 

659 

639 

620 

602 

584 

103 

—14 

824 

796 

770 

745 

721 

698 

677  !  656 

636 

617 

598 

104 

—15 

0.850 

0.820 

0-793 

0.766 

0.741 

0.718 

0.695  !  0.673 

0.652 

0.632 

0.613 

105 

—16 

878 

847 

817 

789 

753 

738 

691 

669 

648 

628 

106 

—17 

908 

875 

-843 

814 

786 

759 

734 

710 

687 

665 

644 

1O7 

-18 

942 

905 

871 

840 

810 

782 

756 

73° 

706 

683 

66  1 

108 

—19 

977 

938 

902 

868 

836 

806 

778 

752 

726 

702 

679 

109 

—2O 

1.017 

0-974 

o-935 

0.898 

0.864 

0.832 

0.802 

0.774 

0.748 

0.722 

0.698 

110 

—21 

—22 

06  1 
x-uo 

1.013 
057 

970 

I.OIO 

966 

894 
927 

860 
890 

828 
856 

7^8 
824 

770 
794 

III 

718 
739 

111 
112 

—23 

166 

106 

°53 

i.  006 

962 

923 

886 

852 

820 

789 

yf 

76l 

113 

—24 

231 

162 

102 

049 

I.OO2 

958 

918 

881 

847 

815 

785 

114 

60° 

59° 

58° 

57° 

56° 

55° 

54° 

53° 

52° 

51° 

5O° 

Polar 

Co-Latitude. 

Dist. 

114 


TABLE  XXX. 


1 

Time-  Azimuths  :   Log  A. 

Latitude. 

Dec. 

46    | 

Polar 
Dist. 

4O° 

41° 

42° 

43 

44° 

45° 

47° 

48°  i   49° 

50° 

o 

35 

8.740 

8.822 

8.892 

8-953 

9.007 

9.056 

9.101 

9.141 

9.179     9-215 

9.248 

o      • 

55 

34 

817 

886 

947 

9.001 

050 

094 

J35 

173 

208        241 

273 

56 

33 

88  1 

941 

995 

044 

088 

129 

167 

202 

235 

266 

295 

57 

32 

936 

989 

9.038 

082 

123 

1  60 

195 

228 

289 

58 

31 

984 

9-032 

076 

117 

154 

189 

222 

253 

282        310 

337 

59 

30 

9.027 

9.071 

9.111 

9.149 

9.184 

9.216 

9.247 

9.276 

9-304     9-330 

9-355 

60 

29 

066 

106 

'43 

178 

211 

241 

270 

298 

324 

349 

373 

61 

28 

IOI 

138 

173 

205 

236 

264 

292 

318 

343 

367 

390 

62 

27 

133 

167 

200 

230 

259 

286 

3I2 

337 

384 

406 

63 

26 

162 

195 

225 

254 

28l 

307 

332 

355 

378        400 

421 

64 

25 

9.190 

9.220 

9.249 

9.276 

9.302 

9-326 

9-35° 

9-373 

9-395 

9.416 

9436 

65 

24 

215 

244 

271 

297 

32I 

345 

367 

389 

410 

43° 

45° 

66 

23 

239 

266 

292 

316 

340 

362 

384           405 

425 

444 

463 

67 

22 

261 

287 

335 

357 

379 

399        4*9 

439 

457 

476 

68 

21 

282 

307 

330 

352 

374 

394 

414 

434 

452 

470 

488 

69 

20 
19 

9.302 
321 

9-325 
343 

9.348 
364 

9385 

9-390 
405 

9.409 
424 

9-429 
442 

9-447 
460 

9-465 
478 

9483 
495 

9.50° 
511 

70 
71 

18 

339 

360 

380 

400 

419 

437 

455 

473 

490 

506 

•    522 

72 

17 

356 

376 

396 

415 

433 

45° 

r 

485 

5O1 

5J7 

S32 

73 

16 

372 

39i 

410 

429 

446 

463 

480 

496 

512 

527 

542 

74 

15 

9-387 

9.406 

9424 

9.442 

9-459 

9.476 

9.492 

9-507 

9-523 

9.538 

9-552 

75 

14 

402 

420 

438 

455 

487 

503 

518 

533 

547 

562 

76 

13 

416 

434 

451 

467 

483 

498 

528 

543 

557 

77 

12 

43° 

447 

463 

479 

494 

5°9 

524 

538 

553 

566 

580 

78 

11 

443 

459 

475 

490 

505 

520 

534 

548 

562 

575 

588 

79 

10 

9-456 

9471 

9487 

9.502 

9.516 

9-53° 

9-544 

9-558 

9-571 

9-584 

9-597 

80 

9 

468 

483 

498 

512 

526 

540 

554 

567 

580 

592 

605 

81 

8 

480 

494 

509 

523 

536 

550 

563 

576 

588 

600 

613 

82 

7 

491 

505 

533 

546 

559 

572 

584 

596 

608 

620 

83 

6 

502 

529 

543 

555 

568 

580 

593 

604 

616 

628 

84 

5 

9-5I3 

9.526 

9-539 

9-552 

9-565 

9-577 

9-589 

9.601 

9.612 

9.624 

9-635 

85 

4 

523 

536 

549 

573 

585 

597 

609 

620 

631 

642 

86 

3 

533 

545 

558 

570 

582 

594 

605 

616 

627 

638 

649 

87 

2 

1 

542 
552 

555 
564 

567 
576 

579 
587 

590 
598 

602 
610 

620 

624 
631 

635 
642 

645 
652 

2 

88 
89 

0 

9.561 

9-573 

9.584 

9-595 

9.606 

9-617 

9.628 

9-638 

9-649 

9.659 

9.669 

90 

—    1 

570 

581 

593 

603 

614 

625 

635 

645 

655 

665 

675 

91 

—   2 

579 

60  1 

611 

622 

632 

642 

652 

662 

672 

68  1 

92 

-   3 

587 

598 

609 

619 

629 

639 

649 

659 

668 

678 

687 

93 

—  4 

595 

606 

616 

626 

636 

646 

656 

665 

675 

684 

693 

94 

—   5 

9.603 

9.614 

9.624 

9-634 

9-643 

9.653 

9.662 

9.672 

9.681 

9.690 

9.699 

95 

—   6 

611 

621 

631 

641 

650 

660 

669 

678 

687 

696 

7°4 

96 

—    7 

619 

629 

638 

648 

657 

666 

675 

684 

693 

701 

710 

97 

—   8 

627 

636 

645 

655 

664 

673 

681 

690 

699 

707 

98 

—  9 

634 

643 

652 

661 

670 

679 

687 

696 

704 

713 

721 

99 

—10 

9.641 

9.650 

9.659 

9.668 

9.676 

9-685 

9-693 

9.702 

9.710 

9.718 

9.726 

1OO 

—11 

648 

657 

666 

674 

683 

691 

699 

707 

715 

723 

731 

101 

—12 

655 

664 

672 

681 

689 

697 

705 

713 

721 

729 

736 

102 

—13 

662 

670 

679 

687 

695 

703 

711 

718 

726 

734 

103 

—14 

668 

677 

685 

693 

701 

708 

716 

724 

731 

739 

746 

1O4 

-15 

9-675 

9-683 

9.691 

9.699 

9.706 

9-7H 

9.722 

9.729 

9-736 

9-744 

9.751 

105 

—16 

68  1 

689 

697 

705 

712 

720 

727 

734 

741 

749 

756 

1O6 

—17 

688 

695 

703 

710 

718 

725 

732 

739 

746 

753 

760 

107 

—18 

694 

701 

709 

716 

723 

73° 

737 

744 

751 

765 

108 

—19 

700 

707 

7H 

721 

729 

736 

742 

.     749 

756 

763 

769 

109 

—20 

9.706 

9-713 

9.720 

9727 

9-734 

9-741 

9-747 

9-754 

9.761 

9.767 

9-774 

110 

—21 

712 

718 

725 

732 

739 

746 

752 

765 

772 

778 

111 

—22 

717 

724 

73  i 

738 

744 

751- 

757 

764 

770 

776 

782 

112 

—23 

723 

736 

743 

749 

756 

762 

768 

774 

780 

787 

113 

—24 

728 

735 

748 

754 

760 

767 

773 

779 

785 

791 

114 

5O° 

49° 

48° 

47° 

46° 

45° 

44° 

43° 

42° 

41° 

40° 

Polar 

Co-Latitude. 

Dist. 

TABLE  XXX. 


115 


Time-  Azimuths:  Log  12. 

Latitude. 

Dec. 

Polar 
Dist. 

40° 

41° 

42° 

43° 

44° 

•45° 

46° 

47° 

48° 

49° 

50° 

35 

0.215 

O.2IO 

0.205 

0.200 

0-195 

0.190 

o.i  86 

0.181 

0.176 

0.171 

0.167 

o 

55 

34 

220 

215 

210 

205 

199 

194 

190 

185 

1  80 

'75 

170 

56 

33 

225 

219 

214 

209 

204 

199 

189 

184 

179 

174 

57 

32 

230 

224 

219 

214 

208 

203 

198 

193 

1  88 

183 

178 

58 

31 

235 

229 

224 

218 

213 

208 

202 

197 

192 

187 

181 

59 

3O 

0.240 

0.234 

0.228 

0.223 

0.217 

0.212 

0.207 

0.201 

0.196 

0.191 

0.185 

60 

29 

245 

239 

233 

228 

222 

216 

211 

205 

200 

195 

189 

61 

28 

250 

244 

238 

232 

227 

221 

215 

210 

2O4 

199 

193 

62 

27 

255 

249 

243 

237 

23I 

225 

22O 

214 

208 

203 

197 

63 

26 

261 

254 

248 

242 

236 

230 

224 

218 

212 

207 

20  1 

64 

25 

0.266 

0.260 

0-253 

0.247 

0.241 

0.235 

0.229 

O.223 

0.217 

0.21  1 

0.205 

65 

24 

272   265 

259 

252 

246 

240 

233 

227 

221 

215 

209 

66 

23 

277   270 

264 

257 

25  I 

244 

238 

232 

226 

220 

213 

67 

22 

283 

276 

269 

263 

256 

249 

243 

236 

230 

224 

218 

68 

21 

289 

282 

275 

268 

26l 

254 

248 

24I 

235 

228 

222 

69 

2O 

0.294 

0.287 

0.280 

0.273 

0.266 

0-259 

0-253 

0.246 

0-239 

0-233 

0.226 

70 

19 

300 

293 

286 

279 

271 

264 

258 

251 

244 

237 

231 

71 

18 

306. 

299 

291 

284 

277 

27O 

263 

256 

249 

242 

235 

72 

17 

312 

305 

297 

290 

282 

275 

268 

261 

254 

247 

240 

73  1 

16 

3*9 

3" 

303 

296 

288 

280 

273 

266 

259 

251 

245 

74  i 

15 

0-325 

0.317 

0.309 

0.301 

0.294 

0-286 

0.278 

O.27I 

0-264 

0.256 

0.249 

75 

14 

332 

323 

315 

307 

299 

292 

284 

276 

269 

26l 

254 

76 

13 

338 

330 

322 

3'3 

305 

297 

289 

282 

274 

266 

259 

77 

12 

345 

336 

328 

3" 

3°3 

295 

287 

279 

271 

264 

78 

11 

352 

343 

334 

326 

3°9 

3OI 

293 

285 

277 

269 

79 

10 

0-359 

0-35° 

0-341 

0-332 

0.324 

0.315 

0.307 

0.298 

0.290 

0.282 

0.274 

80 

9 

366 

357 

348 

339 

330 

321 

313 

3°4 

296 

287 

279 

81 

8 

374 

364 

355 

345 

336 

327 

3*9 

3IO 

301 

293 

285 

82 

7 

381 

371 

362 

352 

343 

334 

325 

316 

3°7 

299 

290 

83 

6 

389 

379 

369 

359 

350 

340 

322 

313 

3°4 

296 

84 

5 

0-397 

0.386 

0.376 

0.366 

o-357 

0-347 

0-338 

0.328 

0.319 

0.310 

0.301 

85 

4 

405 

394 

384 

374 

364 

354 

344 

335 

325 

316 

307 

86 

3 

4i3 

402 

392 

381 

37i 

361 

35  i 

341 

332 

322 

87 

2 

421 

410 

399 

389 

378 

368 

358 

348 

338 

328 

319 

88 

1 

43° 

419 

407 

397 

386 

375 

365 

355 

345 

335 

325 

89 

O 

0-439 

0.427 

0.416 

0.405 

0-394 

0-383 

0.372 

0.362 

0.351 

0.341 

0.331 

90 

.      | 

448 

436 

424 

413 

402 

390 

380 

369 

358 

348 

338 

91 

—  2 

458 

445 

433 

421 

410 

398 

387 

376 

365 

355 

344 

92 

—  3 

467 

455 

442 

43° 

418 

406 

395 

384 

373 

362 

93 

—  4 

477 

464 

451 

439 

427 

415 

403 

380 

369 

358 

94 

—  5 

0.488 

0-474 

0.461 

0.448 

0.436 

0.423 

0.411 

0-399 

0.388 

0.376 

0-365 

95 

—  6 

498 

484 

47i 

457 

445 

432 

420 

396 

384 

372 

96 

—  7 

509 

495 

481 

467 

454 

44i 

428 

416 

404 

392 

380 

97 

—  8 

520 

506 

491 

477 

464 

45° 

437 

424 

412 

400 

387 

98 

—  9 

532 

517 

502 

488 

474 

460 

446 

433 

420 

408 

395 

99 

—10 

0-544 

0.529 

0-513 

0-499 

0.484 

0.470 

0.456 

0.442 

0.429 

0.416 

0.403 

100 

—11 

557 

54i 

525 

5^0 

495 

480 

466 

452 

438 

425 

412 

101 

—  12 

570 

553 

537 

521 

506 

491 

476 

462 

447 

434 

421 

102 

—  13 

584 

566 

549 

533 

517 

502 

486 

472 

457 

443 

429 

103 

—14 

598 

580 

562 

545 

629 

513 

497 

482 

467 

453 

438 

104 

—15 

0.613 

0-594 

0.576 

0.558 

0.541 

0-525 

0.508 

0-493 

0.477 

0.463 

0.448 

1O5 

—  16 

628 

609 

590 

554 

537 

520 

5°4 

488 

473 

458 

106 

—18 
—19 

644 
66  1 
679 

624 
640 
657 

604 
620 
636 

615 

567 
581 
596 

549 
563 
577 

532 
545 
558 

515 

527 
540 

499 
5*0 
522 

483 
494 
505 

468 
478 
489 

107 
108 
109 

-20 
—21 

0.698 
718 

0.675 
693 

0.653 
670 

0.631 
648 

0.611 
626 

°'oo6 

0.572 
586 

0-553 
567 

0-535 
548 

0.517 
530 

0.500 

5*2 

110 
111 

—22 

739 

689 

665 

643 

.621 

§01 

581 

561 

543 

524 

112 

—23 
—24 

761 
785 

734 
756 

729 

684 
703 

660 
679 

638 
655 

616 
633 

595 
611 

575 
590 

556 
570 

537 

113  1 
114  ! 

5O° 

49° 

48° 

47° 

46° 

45° 

44° 

43° 

42° 

41° 

40° 

Dec 

Polar 

Co-Latitude. 

Dist. 

116 


TABLE  XXX. 


Time-  Azimuths:  Log  A. 

Latitude. 

Dec. 

«8> 

6O° 

Polar 
Dist. 

50° 

51° 

52° 

53° 

54° 

55° 

56-       57" 

59 

o 

9.248 

9.280 

9-309 

9-337 

9-364 

9-39° 

9-415 

9-439 

9.462 

9.484 

9.506 

55 

34 

273 

302 

33° 

357 

383 

407 

431 

454 

476 

498 

518 

56 

33 

295 

323 

35° 

376 

400 

424 

447 

468 

490 

510 

530 

57 

32 

343 

369 

393 

417 

439 

461 

482 

503 

523 

542 

58  ] 

31 

337 

362 

386 

410 

432 

454 

475 

495 

534 

553 

59 

30 

9-355 

9.380 

9403 

9.425 

9-447 

9.468 

9.488 

9-508 

9-527 

9-545 

9-564 

60 

29 

373 

396 

419 

440 

461 

481 

5°i 

520 

538 

556 

574 

61 

28 

390 

412 

434 

454 

474 

494 

513 

531 

549 

566 

583 

62 

27 

406 

427 

448 

468 

487 

506 

524 

542 

559 

576 

593 

63 

26 

421 

442 

462 

481 

499 

518 

535 

552 

569 

586 

602 

64 

25 

9436 

9.456 

9-475 

9-493 

9-5U 

9-529 

9-546 

9.562 

9-579 

9-595 

9.611 

65 

24 

45° 

469 

487 

505 

523 

540 

556 

572 

588 

604 

619 

66  i 

23 

463 

481 

499 

533 

55° 

566 

582 

597 

612 

627 

67 

22 

476 

493 

511 

527 

544 

560 

575 

591 

606 

620 

635 

68  ! 

21 

488 

505 

522 

538 

554 

569 

585 

600 

614 

628 

643 

69 

20 

9.500 

9.516 

9-532 

9-548 

9-564 

9-579 

9-594 

9.608 

9.622 

9.636     9.650 

7O 

19 

5" 

527 

543 

558 

573 

588 

602 

616 

630 

644 

657 

71 

18 

522 

537 

553 

567 

582 

596 

610 

624 

638 

651 

664 

72 

17 

532 

547 

562 

577 

605 

618 

632 

645 

658 

6/1 

73 

16 

542 

557 

57i 

586 

599 

613 

626 

639 

652 

665 

677 

74 

15 

9-552 

9.566 

9.580 

9-594 

9.608 

9.621 

9-634 

9.646 

0.659 

9.671 

9-683 

75 

14 

562 

575 

589 

603 

616 

628 

641 

653 

666 

678 

689 

76 

13 
12 

584 
593 

606 

618 

623 
631 

636 
643 

648 
655 

660 
667 

672 
679 

684 
690 

695 
701 

77 

78 

11 

588 

601 

614 

626 

638 

650 

662 

673 

696 

707 

79 

10 

9-597 

9.609 

9.621 

9-633 

9.645 

9-657 

9.668 

9.680 

9.691 

9.702 

9-7I3 

80 

9 

605 

617 

629 

641 

652 

664 

675 

686 

697 

707 

718 

81 

8 

613 

624 

636 

648 

659 

670 

68  1 

692 

702 

713 

723 

82     : 

7 

620 

632 

643 

654 

665 

676 

687 

698 

708 

718 

728 

83  i 

6 

628 

639 

650 

661 

672 

682 

693 

703 

713 

724 

733 

84  | 

5 

9.635 

9.646 

9.657 

9.668 

9.678 

9.688 

9.699 

9-709 

9.719 

9.729 

9-738 

85 

4 

642 

653 

663 

674 

684 

694 

7°4 

7H 

724 

734 

743 

86 

3 

649 

659 

670 

680 

690 

700 

710 

720 

729 

739 

748 

87 

2 

656 

666 

676 

686 

696 

706 

7*5 

725 

734 

743 

753 

88 

1 

662 

672 

682 

692 

702 

711 

721 

730 

739 

757 

89  ! 

0 

9.669 

9.679 

9.688 

9.698 

9.707 

9.717 

9.726 

9-735 

9-744 

9-753 

9.762 

90 

—    1 

675 

685 

694 

703 

713 

722 

731 

74° 

748 

757 

766 

91  ! 

—   2 

681 

691 

700 

7°9 

718 

727 

736 

744 

753 

762 

770 

92 

—   3 

687 

696 

705 

723 

732 

749 

758 

766 

774 

93 

—  4 

693 

702 

711 

720 

728 

737 

745 

754 

762 

770 

778 

94 

—   5 

9.699 

9.708 

9.716 

9-725 

9-733 

9.742 

9-750 

9-758 

9.766 

9-774 

9.782 

95 

—  6 

704 

722 

73° 

738 

747 

755 

763 

771 

778 

786 

96 

—    7 

710 

7i8 

727 

735 

743 

75i 

759 

767 

775 

^ 

790 

97 

—   g 

715 

724 

732 

740 

748 

756 

763 

771 

779 

786 

794 

98  ! 

—   9 

721 

729 

737 

745 

753 

760 

768 

775 

783 

790 

798 

99 

-10 

9.726 

9-734 

9.742 

9.750 

9-757 

9-765 

9.772 

9.780 

9.787 

9-794 

9.801 

100 

1  —11 

73  i 

739 

747 

754 

762 

769 

776 

784 

791 

798 

805 

101 

—12 

736 

744 

75  i 

759 

766 

773 

780 

787 

795 

802 

809 

1O2 

—13 

74  i 

749 

756 

763 

77° 

777 

784 

791 

798 

805 

812 

1O3  ' 

-14 

746 

753 

760 

768 

775 

782 

788 

795 

802 

809 

816 

104  ; 

—15 

9-751 

9-765 

9.772 

9-779 

9.786 

9-792 

9-799 

9.806 

9.812 

9.819 

1O5 

—16 

756 

763 

769 

776 

783 

790 

796 

803 

809 

816 

822 

106 

—17 

760 

767 

774 

780 

787 

794 

800 

807 

813 

819 

825 

107 

-18 

765 

771 

778 

785 

79  i 

797 

804 

810 

816 

823 

829 

1O8  : 

—19 

769 

776 

782 

789 

795 

801 

807 

814 

820 

826 

832 

1O9 

—2O 

9-774 

9.780 

9.786 

9-793 

9-799 

9.805 

9.811 

9.817 

9-823 

9.829 

9.835 

1  10 

—21 

778 

784 

791 

797 

803 

809 

815 

821 

827 

832 

838 

Mi  i 

—22 

782 

789 

795 

801 

807 

812 

818 

824 

830 

836 

841 

112  ! 

-23 

787 

793 

804 

810 

816 

822 

828 

833 

839 

844 

113 

-24 

791 

797 

802 

808 

814 

820 

825 

831 

836 

842 

847 

114 

4O° 

39° 

38° 

37° 

36° 

35° 

34° 

33° 

32° 

31° 

3O° 

! 
Polar  | 

Co-Latitude. 

Dist. 

TABLE  XXX. 


117 


Time-Azimuths:  L>og  !J. 

Latitude. 

Polar 

Dsc. 

50° 

51° 

52° 

53   54° 

55° 

56   57°   58° 

59° 

6O° 

Dist. 

35 

0.167 

0.162 

0.157 

0.153  i  0.148 

0.144 

0.139  i  0.135  0.131 

0.126 

O.I22 

o 

55 

170 

1  66 

161 

156  !  152 

H7 

142  !   138   133 

129 

I25 

56 

33 

174 

169 

164 

1  60 

155 

150 

145   141    136 

132 

127 

57 

32 

178 

173 

1  68 

163 

158 

153 

149 

H4   139 

135 

I30 

58 

31 

181 

176 

171 

1  66 

161 

157 

152 

147   142 

137 

133 

59 

30 

0,185 

0.180 

0.175 

0.170 

0.165 

o.i  60 

0.155 

0.150  j  0.145 

0.140 

0.135 

60 

29 

28 

199 
193. 

184 
188 

179 

182 

173 
177 

1  68 
172 

I3 

167 

158 
161 

'53 
156 

148 

H3 
146 

138 
141 

61 

62 

27 

197 

191 

186 

181 

170 

165 

159 

154 

149 

1  44 

63 

26 

201 

195 

190 

184 

179 

173 

1  68 

163 

157 

152 

*47 

64 

25 

0.205 

0.199 

0.194 

o.i  88 

0.182 

0.177 

0.171 

0.166 

o.i  60 

0.155 

0.150 

65 

24 

209 

203 

198 

192 

1  86 

1  80 

175 

169 

164 

158 

153 

66 

23 

213 

207 

201 

196  1   190 

184 

178 

173 

167 

161 

156 

67 

22 

218 

212 

2O5 

200 

194 

187 

182 

176 

170 

164 

68 

21 

222 

216 

210 

203 

197 

191 

185 

179 

173 

1  68 

162 

69 

20 

0.226 

0.220 

O.2I4 

0.207 

0.201 

0.195 

0.189 

0.183 

0.177 

0.171 

0.165 

70 

19 

231 

224 

218 

211 

205 

199 

193 

186 

1  80 

174 

1  68 

71 

18 

235 

229 

222 

215 

209 

203 

196 

190 

184 

177 

171 

72 

17 

240 

233 

226 

220 

213 

207 

200 

194 

187 

181 

'75 

73 

16 

245 

23I 

224 

217 

210 

2O4 

197 

191 

184 

178 

74 

15 

0.249 

0.242 

0-235 

0.228 

0.221 

O.2I4 

0.208 

0.201 

0.194 

0.188 

0.181 

75 

14 

254 

247 

240 

232 

225 

218 

212 

2O5 

198 

191 

185 

76 

13 

259 

252 

244 

237 

230 

223 

216 

2O9  j   2O2 

195 

188 

77 

12 

264 

256 

249 

24I 

234 

227 

220 

213 

205 

198 

192 

78 

11 

269 

26l 

253 

246 

239 

231 

224 

216 

209 

202 

195 

79 

10 

0.274 

0.266 

0.258 

0.251 

0.243 

0.235 

0.228 

0.220 

0.213 

0.206 

0.199 

80 

9 

279 

271 

263 

255 

248 

240 

232 

225 

217 

210 

202 

81 

8 

•285 

276 

268 

260 

252 

244 

237 

229 

221 

214 

2O6 

82 

7 

29O 

282 

273 

265 

257 

249 

241 

233 

225 

218 

210 

83 

6 

296 

287 

278 

270 

262 

253 

245 

237 

230 

222 

214 

84 

5 

0.301 

0.292 

0.284 

0.275 

0.267 

0.258 

0.250 

0.242 

0.234 

0.226 

0.218 

85 

4 

3 

307 
313 

298 
304 

289 
295 

280 
286 

272 

277 

268 

255 
260 

246 
251 

238 
242 

230 
234 

222 

226 

86 

87 

2 

1 

319 
325 

3°9 
315 

300 
306 

29I 
297 

282 
287 

273 

278 

264 
269 

260 

252 

238 
243 

230 
234 

88 
89 

0 

Q-331 

0.321 

0.312 

0.302 

0.293 

0.283 

0.274 

0.265 

0.256 

0.247 

0.239 

90 

I  —  1 

338 

328 

318 

308 

298 

289 

279 

270 

26l 

252 

243 

91 

''  —  2 

344 

334 

324 

3°4 

294 

285 

275 

266 

257 

247 

92 

—  3 

351 

33° 

320 

3IO 

300 

29O 

28l 

271 

26l 

252 

93 

—  4 

358 

347 

337 

326 

316 

306 

296 

286 

276 

266 

257 

94 

—  5 

0-365 

o-354 

0-343 

0.332 

0.322 

0.312 

0.301 

0.291 

0.28l 

0.271 

0.262 

95 

—  6 

372 

361 

35o 

339 

328 

318 

307 

297 

287 

276 

267 

96 

i  —  7 

380 

368 

357 

346 

335 

324 

313 

302 

292 

282 

272 

97 

—  8 

387 

376 

364 

353 

330 

319 

308 

298 

287 

277 

98 

—  9 

395 

383 

371 

360 

348 

336 

325 

3H 

303 

293 

282 

99 

—10 

0.403 

0.391 

0-379 

0.367 

0-355 

0-343 

0.332 

0.320 

0.309 

0.298 

0.287 

10O 

—11 

412 

399 

386 

374 

362 

350 

338 

327 

3°4 

293 

101 

—12 

421 

407 

394 

382 

369 

357 

345 

333 

32I 

3IO 

299 

1O2 

-13 

429 

416 

402 

390 

377 

364 

352 

340 

328 

3I6 

305 

103 

—14 

438 

425 

411 

398 

384 

372 

359 

347 

334 

322 

310 

104 

-15 

0.448 

0-434 

0.420 

0.406 

0.392 

0.379 

0.366 

0-354 

0.341 

0.329 

0-317 

105 

—  16 

458 

443 

429 

414 

401 

387 

374 

361 

348 

335 

323 

106 

!  —17 

468 

453 

438 

423 

409 

395 

382 

368 

355 

342 

329 

107 

i  —18 

478 

463 

447 

433 

418 

404 

390 

376 

362 

349 

336 

108 

—19 

489 

473 

457 

442 

427 

412 

398 

384 

370 

356 

343 

1O9 

—20 

0.500 

0.484 

0.468 

0.452 

0.436 

0.421 

0.406 

0.392 

0.378 

0.364 

0-35° 

110 

—21 

512 

495 

478 

462 

446 

43  * 

415 

400 

386 

372 

357 

111 

—22 

524 

507 

489 

473 

456 

440 

425 

409 

394 

380 

365 

112 

—23 

537 

519 

484 

467 

45° 

434 

418 

403 

388 

373 

113 

—24 

550 

53i 

513 

495 

478 

461 

444 

428 

412 

396 

114 

4O° 

39° 

38° 

37° 

36° 

35° 

34°   33° 

32° 

31° 

30° 

Polar 

Co-Latitude. 

Dist. 

118 


TABLE  XXX. 


Time-Azimuths:  JLog  A. 

Latitude. 

_ 

Polar 

60° 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

70° 

Dist. 

35 

9.506 

9-527 

9-547 

9-567 

9.586 

9.605 

9-623 

9.641 

9-659 

9-677 

9-694 

55 

34 

518 

538 

558 

577 

596 

614 

632 

650 

667 

684 

701 

56 

33 

530 

569 

587 

606 

623 

641 

658 

675 

691 

707 

57 

32 

542 

56i 

579 

597 

615 

632 

649 

666 

682 

698 

58 

31 

553 

589 

606 

624 

640 

657 

673 

689 

705 

720 

59 

30 

9-564 

9.581 

9-598 

9.615 

9.632 

9.648 

9-665 

9.680 

9.696 

9.711 

9.726 

60 

29 

574 

59i 

607 

624 

640 

656 

672 

687 

702 

717 

732 

61 

28 

583 

600 

616 

632 

648 

664 

679 

694 

709 

723 

737 

62 

27 

593 

609 

625 

640 

656 

671 

686 

700 

729 

743 

63 

26 

602 

618 

633 

648 

663 

678 

692 

706 

721 

734  !      748 

64 

25 

9.611 

9.626 

9.641 

9.656 

9.670 

9-685 

9-699 

9.712 

9.726 

9.740  j  9.753 

65 

24 

619 

634 

649 

663 

677 

691 

705 

718 

732 

745 

758 

66 

23 
22 

627 
635 

642 
649 

656 
663 

670 
677 

684 
690 

697 
703 

711 

717. 

724 
73° 

737 
742 

750 

755 

i 

67 
68 

21 

643 

656 

670 

683 

696 

709 

722 

735 

747 

760 

772 

69 

2O 

9.650 

9-663 

9-677 

9.690 

9.702 

9715 

9.728 

9.740 

9-752 

9-765 

9-776 

7O 

19 
18 

657 
664 

670 
677 

689 

696 
702 

708 
714 

721 

726 

733 
738 

745 
75° 

III 

769 
773 

781 
785 

71 

72 

17 
16 

677 

683 
689 

695 
701 

708 
713 

720 

725 

737 

743 
748 

760 

766 

778 
782 

789 
793 

73 
74 

15 

9.683 

9.695 

9.707 

9.719 

9-73° 

9.742 

9-753 

9.764 

9-775 

9.786 

9-797 

75 

14 

689 

701 

713 

724 

735 

747 

769 

779 

790 

801 

76 

13 

695 

707 

729 

740 

762 

773 

783 

794 

804 

77 

12 

701 

713 

724 

734 

745 

756 

767 

777 

787 

798 

808 

78 

11 

707 

718 

729 

739 

750 

760 

781 

79  i 

80  1 

811 

79 

10 

9-7I3 

9-723 

9-734 

9-744 

9-755 

9765 

9-775 

9.785 

9-795 

9.805 

9.815 

80 

9 

718 

728 

739 

749 

759 

769 

779 

789 

799 

809 

.  818 

81 

8 

723 

733 

744 

754 

764 

774 

783 

793 

803 

812 

821 

82 

7 

728 

738 

748 

758 

768 

778 

787 

797 

806 

815 

825 

83 

6 

733 

743 

753 

763 

772 

782 

791 

800 

810 

819 

828 

84 

5 

9-738 

9.748 

9758 

9-767 

9.776 

9.786 

9-795 

9.804 

9-813 

9.822 

9-831 

85 

4 

743 

753 

762 

771 

780 

789 

798 

807 

816 

825 

834 

86 

3 

748 

757 

766 

775 

793 

802 

811 

819 

828 

837 

87 

2 

753 

761 

771 

779 

788 

797 

806 

814 

823 

831 

840 

88 

1 

757 

766 

775 

783 

792 

801 

809 

818 

826 

834 

842 

89 

0 

9.762 

9.770 

9-779 

9.787 

9.796 

9.804 

9.813 

9.821 

9.829 

9.837 

9-845 

90 

—   1 

766 

774 

783 

791 

799 

808 

816 

824 

832 

840 

848 

91 

1  —   2 

770 

778 

787 

795 

803 

811 

819 

827 

835 

843 

851 

92 

—   * 

774 

782 

790 

799 

807 

815 

822 

830 

838 

846 

853 

93 

I  —  4 

778 

786 

794 

802 

810 

818 

826 

833 

841 

848 

856 

94 

—   5 

9.782 

9.790 

9.798 

9.806 

9.813 

9.821 

9.829 

9.836 

9.844 

9.851 

9-858 

95 

[  __  ** 

786 

794 

802 

809 

817 

824 

832 

839 

846 

854 

861 

96 

—   7 

790 

798 

805 

813 

820 

827 

835 

842 

849 

856 

863 

97 

—   8 

794 

801 

809 

816 

823 

830 

838 

845 

852 

859 

866 

98 

—  9 

798 

805 

812 

819 

826 

833 

841 

848 

854 

861 

868 

99 

—  1O 

9.801 

9.809 

9-815 

9.823 

9.830 

9.836 

9-843 

9.850 

9.857 

9.864 

9.871 

1OO 

—11 

805 

812 

819 

826 

833 

839 

846 

853 

8f)0 

866 

873 

1O1 

—12 

809 

815 

822 

829 

836 

842 

849 

856 

862 

869 

875 

102 

-13 
—14 

812 
816 

819 
822 

825 
829 

832 
835 

839 
842 

845 
848 

852 
854 

858 
861 

865 
867 

871 
873 

877 
880 

103 
104 

-15 

9.819 

9.825 

9.832 

9.838 

9-845 

9.851 

9-857 

9-863 

9.869 

9.876 

9.882 

105 

—16 

822 

828 

835 

841 

847 

854 

860 

866 

872 

878 

884 

106 

—17 

825 

832 

838 

844 

850 

856 

862 

868 

874 

880 

886 

1O7 

—18 

829 

835 

841 

847 

853 

859 

865 

871 

877 

882 

888 

1O8 

—19 

832 

838 

844 

850 

856 

862 

867 

873 

879 

885 

890 

109 

—20 

9.835 

9.841 

9.847 

9.853 

9-858 

9.864 

9.870 

9-875 

9.881 

9.887 

9.892 

no 

—21 

838 

844 

850 

855 

86  1 

867 

872 

878 

883 

889 

894 

in 

—22 

841 

847 

852 

858 

864 

869 

875 

880 

886 

891 

896 

112 

—23 

844 

850 

855 

86  1 

866 

872 

877 

882 

888 

893 

898 

113 

—24 

847 

853 

858 

864 

869 

874 

879 

885 

890 

895 

900 

114 

30° 

29° 

28° 

27° 

26° 

25° 

24° 

23° 

22° 

21° 

20° 

i 

Dec. 

Polar 
Dist.  i 

Co-Latitude. 

TABLE  XXX. 


119 


Time-  Azimuths:  Log  B. 

Latitude. 

Polar 

Dec. 

6O° 

61° 

62° 

63° 

64° 

65° 

66° 

67° 

68° 

69° 

70° 

Dist. 

o 

35 

0.122 

0.118 

0.113 

0.109 

0.105 

O.IOI 

0.097 

0.092 

0.088 

0.084 

0.080 

55 

34 

125 

120 

116 

in 

107 

103 

099 

094 

090 

086 

082 

56 

33 

127 

123 

118 

114 

109 

105 

101 

096 

092 

088 

083 

57 

32 

130 

125 

121 

116 

TI2 

107 

103 

098 

094 

090 

085 

58 

31 

»33 

128 

I23 

119 

114 

110 

105 

100 

096 

091 

087 

59 

30 

0.135 

0.131 

0.126 

O.I2I 

O.II7 

O.II2 

0.107 

0.103 

0.098 

0.093 

0.089 

60 

29 

138 

*33 

129 

124 

119 

114 

109 

105 

100 

095 

091 

61 

28 

141 

136 

131 

126 

121 

116 

112 

107 

102 

097 

092 

62 

27 

144 

139 

J34 

129 

124 

119 

114 

109 

IO4 

099 

094 

63 

26 

H7 

142 

136 

'31 

126 

121 

116 

in 

106 

101 

096 

64 

25 

0.150 

0.144 

0.139 

0.134 

0.129 

0.124 

0.118 

0.113 

0.108 

0.103 

0.098 

65 

24 

153 

H7 

142 

136 

131 

126 

121 

115 

no 

105 

IOO 

66 

23 

I56 

!5° 

H5 

139 

134 

128 

123 

118 

112 

107 

102 

67 

22 

159 

153 

147 

142- 

I36 

131 

125 

1  20 

114 

109 

I04 

68 

21 

156 

15° 

H5 

139 

'33 

128 

122 

117 

in 

106 

69 

20 

0.165 

0.159 

°-I53 

0.147 

0.142 

0.136 

0.130 

0.125 

0.119 

0.113 

0.108 

70 

19 

1  68 

162 

156 

150 

144 

138 

133 

127 

121 

115 

no 

71 

18 

171 

165 

J59 

*53 

H7 

141 

135 

129 

123 

118 

112 

72 

17 

*75 

168 

162 

156 

^o 

144 

138 

132 

126 

120 

114 

73 

16 

178 

171 

165 

159 

153 

146 

140 

134 

128 

122 

116 

74 

15 

0.181 

o.i75 

o.i  68 

0.162 

0-155 

0.149 

0.143 

0.137 

O.I3I 

O.I24 

0.118 

75 

14 

185 

178 

171 

165 

I58 

!52 

146 

139 

133 

127 

120 

76 

13 

iSS 

181 

'  175 

168 

161 

155 

148 

142 

135 

I29 

123 

77 

12 

192 

185 

178 

171 

164 

158 

151 

144 

138 

131 

125 

78 

11 

195 

1  88 

181 

174 

167 

161 

154 

H7 

140 

134 

127 

79 

10 

0.199 

0.191 

0.185 

0.177 

0.170 

0.163 

0.157 

0.150 

0.143 

0.136 

0.129 

8O 

9 

202 

195 

1  88 

181 

174 

1  66 

159 

153 

I46 

I39 

I32 

81 

8 

206 

199 

191 

184 

177 

170 

162 

155 

148 

I4I 

'34 

82 

7 

2IO 

202 

J95 

187 

1  80 

J73 

165 

151 

144 

137 

83 

6 

214 

206 

198 

191 

183 

176 

1  68 

161 

154 

146 

139 

84 

5 

0.2  1  8 

O.2IO 

0.202 

0.194 

0.187 

o.i79 

0.171 

0.164 

0.156 

0.149 

0.142 

85 

4 

222 

214 

206 

198 

190 

182 

•174 

167 

159 

152 

144 

86 

3 

226 

218 

2IO 

201 

193 

185 

178 

170 

154 

147 

87 

2 

230 

222 

2I3 

205 

197 

189 

181 

173 

165 

157 

149 

88 

1 

234 

226 

217 

209 

2OI 

192 

184 

i'/6 

168 

1  60 

152 

89 

O 

0.239 

0.230 

0.221 

O.2I3 

0.204 

0.196 

0.188 

0.179 

0.171 

0.163   0-155 

90 

—  1 

243 

234 

225 

217 

208 

199 

191 

183 

174 

1  66 

158 

91 

—  2 

247 

239 

229 

221 

212 

203 

194 

-  1  86 

177 

169 

1  60 

92 

—  3 

252 

243 

234 

225 

216 

207 

198 

189 

181 

172 

l63 

93 

—  4 

257 

247 

238 

229 

220 

211 

202 

193 

184 

175 

1  66 

94 

-  5 

0.262 

0.252 

O.242 

0.233 

O.224 

0.214 

O.2O5 

0.196 

0.187 

0.178 

0.169 

95 

—  6 

267 

257 

247 

237 

228 

218 

209 

200 

190 

181 

172 

96 

—  7 

272 

262 

252 

242 

232 

222 

213 

203 

194 

185 

*75 

97  ! 

—  8 

277 

267 

256 

246 

236 

226 

217 

207 

197 

188 

179 

98 

—  9 

282 

272 

26l 

251 

24I 

23I 

221 

211 

201 

191 

182 

99  i 

—10 

0.287 

0.277 

0.266 

0.256 

0.245 

0-235 

0.225 

O.2I5 

0.205 

o.i95 

0.185 

100 

—11 

293 

282 

271 

26l 

250 

239 

229 

219 

209 

199 

189 

101  ! 

—12 

299 

287 

276 

266 

255 

244 

233 

223 

213 

202 

192 

102 

—13 

3°5 

293 

282 

271 

260 

249 

238 

227 

217 

206 

196 

103 

—14 

310 

299 

287 

276 

265 

253 

242 

231 

221 

210 

199 

104 

—15 

0.317 

0.305 

0.293 

0.28l 

0.270 

0.258 

0.247 

0.236 

0.225 

0.214 

0.203 

1O5 

—16 

323 

3" 

299 

287 

275 

263 

252 

240 

229 

218 

207 

106 

—17 

329 

3i7 

3°5 

292 

280 

269 

257 

245 

234 

222 

211" 

107 

-18 

336 

323 

311 

298 

286 

274 

262 

250 

238 

227 

215 

108 

—19 

343 

330 

3i7 

3°4 

292 

279 

267 

255 

243 

231 

219 

109 

-20 

o-35o 

0-337 

0.324 

0.310 

0.298 

0.285 

0.272 

0.260 

0.248 

0.235 

0.224 

110  ! 

—21 

357 

344 

33° 

3*7 

3°4 

291 

278 

265 

253 

240 

228 

Ill 

—22 

365 

351 

337 

323 

3IO 

297 

283 

270 

258 

245 

232 

112 

—23 
—24 

373 
381 

358 
366 

344 

35  i 

330 
337 

316 
323 

3°3 
3°9 

289 
295 

276 
282 

263 
268 

250 
255 

237 
242 

113 
114 

Ti*»r« 

30° 

29° 

28° 

27° 

26° 

25° 

24° 

23° 

22° 

21° 

2O° 

Polar 

uec. 

Co-Latitude. 

Dist. 

120 


TABLE  XXX. 


Time-Azimuths:  Log  A. 

Latitude. 

_ 

Polar 

70° 

71° 

72° 

73D 

74° 

75° 

76° 

77° 

78° 

79° 

8O° 

Dist. 

35 

9.694 

9.711 

9.727 

9-743 

9.760 

9-775 

9.791 

9.807 

9.822 

9.838 

9.853 

55 

34 

701 

717 

733 

749 

765 

780 

796 

811 

826 

841 

856 

56 

33 
32 

707 

723 
729' 

739 
745 

£ 

770 
775 

785 
790 

800 
804 

815 
819 

830 
833 

844 
848 

859 
862 

57 

58 

31 

720 

735 

750 

765 

780 

794 

808 

823 

837 

851 

865 

59 

30 

9.726 

9.741 

9-755 

9.770 

9.784 

9.798 

9.812 

9.826 

9.840 

9-854 

9.867 

60 

29 

28 

732 
737 

746 
752 

761 
766 

775 
779 

789 
793 

802 
806 

816 
820 

830 
833 

843 
846 

& 

870 
873 

61 

62 

26 

743 

748 

III 

770 

775 

784 

788 

810 
814 

824 
827 

837 
840 

849 
852 

862 
865 

875 

878 

63 
64 

25 

9-753 

9.766 

9.779 

9.792 

9.805 

9.818 

9.830 

9-843 

9.855 

9.868 

9.880 

65 

24 

771 

796 

809 

821 

834 

846 

858 

870 

882 

66 

23 
22 
21 

i 

772 

jg 

784 

792 
796 

800 
804 
808 

813 
816 
820 

825 
828 
831 

837 
840 

843 

849 
852 

855 

861 
866 

873 

875 
878 

885 
887 
889 

67 

68 
69 

20 

9.776 

9.788 

9.800 

9.812 

9-823 

9.835 

9.846 

9-857 

9.869 

9.880 

9.891 

70 

19 

781 

792 

804 

815 

827 

838 

849 

860 

872 

882 

893 

71 

18 

785 

796 

807 

819 

830 

841 

852 

863 

874 

884 

895 

72 

17 

789 

800 

Bn 

822 

833 

844 

854 

865 

876 

886 

897 

73 

16 

793 

804 

815 

826 

836 

847 

857 

868 

878 

888 

899 

74 

15 

9-797 

9.808 

9.818 

9.829 

9-839 

9.850 

9.860 

9.870 

9.880 

9.890 

9.900 

75 

14 

801 

8n 

822 

832 

842 

852 

862 

872 

882 

892 

902 

76 

13 
12 

804 
808 

815 

818 

828 

835 
838 

845 
848 

855 
857 

865 
867 

875 
877 

884 

887 

896 

904 
906 

77 

78 

11 

811 

821 

831 

841 

850 

860 

870 

879 

889 

898 

907 

79 

10 

9.815 

9.825 

9-834 

9.844 

9-853 

9.863 

9.872 

9.881 

9.891 

9.900 

9.909 

8O 

'9 

818 

828 

837 

846 

856 

865 

874 

883 

893 

902 

911 

81 

8 
7 

821 
825 

831 
834 

840 
843 

849 

852 

858 
861 

867 
870 

877 
879 

885 
887 

gg 

903 
9°5 

912 
914 

82 
83 

6 

828 

837 

846 

855 

863 

872 

881 

889 

898 

907 

915 

84 

5 
4 

9.831 
834 

*f£ 

9ifi 

**& 

9.866 
868 

9-874 
877 

9885 

9.891 
893 

9.900 
902 

9.908 
910 

9.917 
918 

85 
86 

3 
2 

837 
840 

ss 

856 

862 
865 

871 

873 

879 
881 

887 
889 

895 
897 

903 
905 

911 
913 

920 
921 

87 
88 

1 

842 

851 

859 

867 

875 

883 

891 

899 

907 

923 

89 

o 

9-845 

9.853 

9.861 

9.869 

9.877 

9.885 

9-893 

9.901 

9.908 

9.916 

9.924 

90 

—    1 

848 

856 

864 

872 

879 

887 

895 

902 

910 

918 

925 

91 

1  —   3 

$1 

X 

866 
868 

$ 

88  1 
883 

889 
891 

897 
898 

904 
906 

912 
913 

919 
920 

926 
928 

92 
93 

!  —  4 

856 

863 

871 

878 

885 

893 

900 

907 

915 

922 

929 

94 

1  —   5 

9.858 

9.866 

9-873 

9.88o 

9.887 

9.895 

9.902 

9.909 

9.916 

9-923 

9.93° 

95 

'  —  6 

86  1 

868 

875 

882 

889 

897 

9°4 

911 

918 

925 

932 

96 

!  —   7 

863 

871 

878 

884 

891 

898 

905 

912 

919 

926 

933 

97 

i  —  8 

866 

873 

880 

886 

893 

900 

907 

914 

920 

927 

934 

98 

—  9 

868 

875 

882 

888 

895 

902 

909 

915 

922 

928 

935 

99 

!  —10 

9.871 

9.877 

9.884 

9.890 

9.897 

9.904 

9.910 

9.917 

9-923 

9-93° 

9.936 

100 

!  —11 

873 

879 

886 

892 

899 

905 

912 

918 

925 

93  i 

937 

101 

1  —12 

875 

882 

888 

894 

901 

907 

9i3 

920 

926 

932 

938 

102 

—13 

fo77 

884 

890 

896 

903 

909 

915 

921 

927 

933 

939 

103 

—14 

880 

886 

892 

898 

904 

910 

916 

922 

929 

935 

940 

104 

-15 

9.882 

9.888 

9.894 

9.900 

9.906 

9.912 

9.918 

9.924 

9-930 

9.942 

105 

—16 

884 

890 

896 

902 

908 

914 

919 

925 

931 

937 

943 

106 

i  —  1"^ 

886 

892 

898 

904 

909 

915 

921 

927 

932 

938 

944 

107 

—18 

888 

894 

900 

9°5 

911 

917 

922 

928 

934 

939 

945 

108 

-19 

890 

896 

901 

907 

913 

918 

924 

929 

935 

940 

946 

109 

—20 

9.892 

9.898 

9.903 

9.909 

9.914 

9.920 

9-925 

9-931 

9-936 

9.941 

9-947 

110 

i  —21 

894 

900 

905 

910 

916 

921 

927 

932 

937 

942 

948 

111 

—22 

896 

902 

907 

912 

917 

923 

928 

933 

938 

943 

949 

112 

—23 

898 

903 

909 

914 

919 

924 

929 

934 

939 

945 

95° 

113 

—24 

900 

905 

910 

915 

921 

926 

93  1 

936 

941 

946 

95  i 

114 

20° 

19°  !   18° 

17° 

16° 

15° 

14° 

13° 

12° 

11° 

10° 

n  r» 

i 

Polar 

Co-Latitude. 

Dist. 

TABLE  XXX. 


121 


Time-Azimuths:  Log  B. 

Latitude. 

Polar 

Dec. 

703 

71° 

72° 

73° 

74° 

75° 

76° 

77° 

78° 

79° 

80° 

Dist. 

35 

0.080 

0.076 

0.072 

0.068 

0.064 

0.060 

0.056 

0.052 

0.048 

0.044 

0.040 

O    | 

55 

34 

082 

078 

073 

069 

065 

06  1 

057 

053 

049 

044 

040 

56 

33 
32 

083 

085 

079 

081 

075 
076 

071 
072 

066 
068 

062 

064 

058 
059 

054 
055 

050 

045 
046 

041 
042 

57 

58 

31 

087 

082 

078 

074 

069 

065 

060 

056 

052 

047 

043 

59 

30 

0.089 

0.084 

0.079 

0.075 

0.071 

0.066 

0.062 

0.057 

°-°53 

0.048 

0.044 

6t> 

29 

091 

086 

08  1 

076 

072 

067 

063 

058 

054 

049 

°45 

61 

28 

092 

088 

083 

078 

073 

069 

064 

060 

055 

050 

046 

62 

094 

089 

085 

080 

075 

070 

065 

06  1 

056 

051 

047 

63 

26 

096 

091 

086 

08  1 

076 

072 

067 

062 

057 

052 

048 

64  , 

25 

0.098 

0.093 

0.088 

0.083 

0.078 

0.073 

0.068 

0.063 

0.058 

0-053 

0.048 

65  i 

24 

IOO 

095 

090 

085 

079 

074 

069 

064 

059 

054 

049 

66  i 

23 

102 

°97 

091 

086 

08  1 

076 

071 

066 

060 

055 

050 

67  ! 

22 

104 

098 

093 

088 

083 

077 

072 

067 

062 

056 

051 

68 

21 

1  06 

IOO 

095 

090 

084 

079 

073 

068 

063 

057 

052 

69 

2O 

o.i  08 

O.I  O2 

0.097 

0.091 

0.086 

0.080 

0.075 

0.069 

0.064 

0.059 

0-053 

70 

19 

no 

104 

099 

°93 

087 

082 

076 

071 

065 

060 

°54 

71 

112 

1  06 

IOO 

°95 

089 

083 

078 

072 

066 

061 

055 

73  i 

17 

114 

1  08 

1  02 

096 

091 

085 

079 

073 

068 

062 

056 

73 

16 

116 

no 

104 

098 

092 

087 

08  1 

075 

069 

063 

057 

74 

15 

0.118 

O.I  12 

o.i  06 

O.IOO 

0.094 

0.088 

0.082 

0.076 

0.070 

0.064 

0.058 

75  ! 

14 

120 

114 

1  08 

1  02 

096 

090 

084 

078 

071 

065 

059 

76 

13 

I23 

116 

110 

104 

097 

091 

085 

079 

073 

067 

060 

77  : 

12 

125 

118 

112 

106 

099 

093 

087 

080 

074 

068 

062 

78  i 

11 

127 

121 

114 

108 

101 

095 

088 

082 

075 

069 

063 

79  | 

10 

0.129 

0.123 

0.116 

O.IIO 

0.103 

0.096 

0.090 

0.083 

0.077 

0.070 

0.064 

8O 

9 

132 

I25 

118 

112 

i°5 

098 

091 

085 

078 

072 

065 

Nl 

127 

120 

114 

107 

IOO 

093 

086 

080 

073 

066 

82  i 

7 

137 

129 

122 

116 

109 

102 

095 

088 

08  1 

074 

067 

83 

6 

139 

132 

125 

118 

in 

I03 

096 

089 

082 

075 

068 

84  ; 

5 

0.142 

0.134 

O.I27 

0.120 

0.113 

0.105 

0.098 

0.091 

0.084 

0.077 

0.070 

85 

4 

144 

137 

129 

122 

115 

107 

IOO 

093 

085 

078 

071 

86 

3 

147 

139 

I32 

124 

117 

IO9 

102 

094 

087 

080 

072 

87 

2 

149 

142 

*34 

126 

119 

III 

I03 

096 

088 

08  1 

074 

88  ; 

1 

152 

144 

136 

128 

121 

113 

105 

098 

090 

082 

075 

89 

0 

0-155 

0.147 

0.139 

0.131 

0.123 

O.II5 

0.107 

0.099 

0.092 

0.084 

0.076 

90 

—  I 

158 

149 

141 

133 

125 

117 

IO9 

101 

093 

085 

077 

91  ! 

—  2 

1  60 

144 

135 

127 

119 

III 

103 

095 

087 

079 

92 

—  3 

163 

155 

146 

138 

I30 

121 

113 

105 

097 

089 

080 

93  ! 

—  4 

166 

158 

149 

140 

I32 

I23 

"5 

107 

098 

090 

082 

94 

—  5 

0.169 

o.i  60 

0.152 

0.143 

0.134 

O.I26 

O.II7 

0.109 

O.IOO 

0.092 

0.083 

95 

-  6 

172 

163 

154 

145 

137 

128 

119 

in 

102 

093 

085 

96 

-  7 

175 

166 

157 

T48 

139 

I30 

121 

113 

104 

095 

086 

97  ! 

—  8 

179 

169 

1  60 

151 

142 

123 

1  06 

097 

088 

98  ; 

—  9 

182 

172 

163 

154 

144 

135 

126 

117 

108 

098 

089 

99 

—10 

0.185 

0.176 

0.166 

0.156 

0.147 

0.137 

0.128 

0.119 

0.109 

O.IOO 

0.091 

100 

—11 

189 

179 

169 

159 

150 

140 

I30 

121 

in 

1  02 

.  093 

101 

—12 

192 

182 

172 

162 

M3 

123 

113 

104 

094 

102 

—13 

196 

185 

175 

165 

155 

13S 

125 

116 

106 

096 

103 

—14 

199 

189 

178 

168 

158 

148 

128 

118 

1  08 

098 

104 

—15 

0.203 

0.192 

0,182 

0.171 

0.161 

0.150 

0.140 

0.130 

O.I  2O 

O.IIO 

O.IOO 

105  ! 

—16 

207 

196 

185 

174 

164 

153 

143 

I32 

122 

112 

101 

1O6 

-17 

211 

200 

189 

178 

167 

156 

146 

135 

I24 

114 

103 

107 

—18 

215 

2O4 

192 

181 

170 

148 

137 

127 

116 

I05 

108 

-19 

219 

208 

196 

185 

173 

162 

I5I 

140 

129 

118 

107 

109 

-20 

0.224 

O.2I2 

0.200 

0.188 

0.177 

0.165 

0.154 

0.143 

O.I3I 

O.I2O 

0.109 

110 

—21 

228 

216 

204 

192 

1  80 

169 

H5 

134 

122 

in 

111 

-22 

232 

220 

208 

196 

184 

172 

1  60 

I48 

I36 

"5 

"3 

112 

—23 

237 

224 

212 

200 

187 

175 

163 

139 

127 

115 

113 

—24 

242 

229 

216 

203 

191 

179 

166 

154 

142 

J30 

118 

114 

1 
i  Dec. 

20° 

19° 

18° 

17° 

16° 

15° 

14° 

13° 

12° 

11° 

10 

Polar 

Diat 

I 

Co-Latitude. 

JLJl&lt* 

122 


TABLE  XXX. 


Time-Azimuths:  JLog:  A. 

Declination. 

T.afr 

Co- 

JJcll. 

—24° 

-25° 

—  26D 

—27 

—28° 

—29° 

—30° 

-31° 

-32° 

—33° 

—34° 

Lat. 

o 

0 

9.328 

9-346 

9-363 

9-38o 

9-397 

9-4I3 

9.428 

9-443 

9-457 

9.471 

9-485 

o 

9O 

1 

344 

362 

379 

395 

411 

426 

441 

455 

469 

483 

496 

89 

2 

360 

377 

393 

409 

424 

439 

453 

467 

481 

494 

5°7 

88 

3 

375 

392 

407 

423 

437 

452 

465 

479 

492 

5°5 

5l8 

87 

4 

390 

406 

421 

436 

45° 

464 

477 

490 

503 

516 

528 

86 

5 

9.404 

9.420 

9-434 

9.448 

9.462 

9.476 

9.489 

9.501 

9-5H 

9.526 

9.538 

85  ; 

6 

418 

433 

447 

461 

474 

487 

500 

512 

524 

536 

547 

84  i 

7 

43i 

446 

459 

472 

485 

498 

5io 

522 

534 

545 

556 

83 

8 

444 

458 

47i 

484 

497 

509 

52i 

532 

544 

555 

566 

82  I 

9 

457 

470 

483 

495 

508 

519 

53i 

542 

553 

564 

575 

81 

10 

9.469 

9.482 

9.494 

9.506 

9.5i8 

9-529 

9-541 

9-552 

9.562 

9-573 

9-584 

wo 

11 

481 

493 

505 

517 

528 

539 

550 

56i 

57i 

582 

592 

79  ' 

1*2 

492 

5°4 

516 

527 

538 

549 

560 

570 

580 

590  j      doo 

7£ 

13 

503 

5'5 

526 

537 

548 

558 

569 

589 

599  i      608 

77  ! 

14 

5H 

525 

536 

547 

557 

568 

578 

588 

597 

607         616 

76 

15 

9o25 

9-536 

9.546 

9-557 

9-567 

9-577 

9.586 

9-596 

9.605 

9.615  i    9.624 

75 

16 

535 

546 

556 

566 

576 

586 

595 

604 

613 

623 

631 

74 

17 

545 

555 

5<>5 

575 

585 

594 

603 

612 

621 

630 

639 

73 

IN 

555 

5&5 

575 

584 

594 

603 

612 

620 

629 

638 

646 

7:2 

19 

564 

574 

584 

593 

602 

611 

620 

628 

637 

645 

653 

71 

20 

9-574 

9-583 

9-593 

9.601 

9.610 

9.619 

9.628 

9.636 

9.644 

9.652 

9.660 

70 

21 

5*3 

592 

602 

610 

618 

627 

635 

643 

652 

659 

667 

69 

22 

592 

601 

610 

618 

626 

635 

643 

651 

659 

666         674. 

6  « 

23 

631 

609 

618 

626 

634 

642 

650 

658 

666 

673 

68  1 

67  : 

24 

609 

618 

626 

634 

642 

650 

658 

665 

673 

680 

687 

66 

25 

9.618 

9.626 

9-634 

9.642 

9.650 

9-657 

9.665 

9.672 

9.680 

9.687 

9.694 

T 

65 

26 

626 

634 

642 

649 

657 

664 

672 

679 

686 

693 

700 

61 

27 

634 

642 

649 

657 

664 

672 

679 

686 

693 

700 

706 

63 

28 

642 

650 

657 

664 

672 

679 

686 

692 

699 

706 

712 

62 

29 

650 

657 

664 

672 

679 

686 

692 

699 

706 

712 

718 

61 

30 

9.658 

9.665 

9.672 

9.679 

9.686 

9.692' 

9.699 

9.706 

9.712 

9.718 

9.724 

60 

31 

655 

672 

679 

6^6 

692 

699 

705 

712 

718 

724 

73° 

59 

32 

673 

679 

686 

693 

699 

706 

712 

7i8 

724 

73° 

736 

5£ 

33 

6So 

687 

693 

700 

706 

712 

718 

724 

73° 

736 

742 

57 

34 

687 

694 

700 

706 

712 

718 

724 

730 

736 

742 

748 

56  j 

35 

9.694 

9.701 

9.707 

9-713 

9.719 

9.725 

9-731 

9-736 

9.742 

9.748 

9-753 

55  1 

36 

701 

708 

713 

719 

725 

73i 

737 

742 

748 

753 

759 

54 

37 

708 

7H 

720 

726 

73i' 

737 

743 

748 

753 

759 

764 

53 

3§ 

715 

721 

727 

732 

738 

743 

749 

754 

759 

764 

769 

52 

39 

722 

727 

733 

738 

744 

749 

754 

760 

765 

770 

775 

51 

40 

9.728 

9-734 

9-739 

9-745 

9-750 

9-755 

9.760 

9-765 

9.771 

9-775 

9.780 

50 

41 

735 

740 

746 

7Si 

756 

761 

766 

771 

776 

781 

785 

49 

42 

74i 

747 

752 

757 

762 

767 

772 

776 

781 

786 

790 

48 

43 

748 

753 

758 

763 

768 

772 

777 

782 

786 

791 

795 

47 

44 

754 

759 

704 

769 

773 

778 

783 

787 

792 

796 

800 

46 

45 

9.760 

9-765 

9.770 

9-775 

9-779 

9.784 

9.788 

9.792 

9-797 

9.801 

9.805 

4,5 

46 

767 

.      77i 

776 

780 

785 

789 

793 

798 

802 

806 

810 

44 

47 

773 

777 

782 

786 

790 

795 

799 

803 

807 

811 

8i5 

43 

48 

779 

783 

788 

792 

796 

800 

804 

808 

812 

816 

820 

42 

49 

785 

789 

793 

797 

801 

805 

809 

813 

817 

821 

825 

41 

50 

9.791 

9-795 

9-799 

9-803 

9.807 

9.811 

9.815 

9.818 

9.822 

9.826 

9.830 

4O 

51 

797 

801 

804 

808 

812 

816 

820 

823 

827 

831 

8.S4 

39 

52 

802 

806 

810 

814 

818 

821 

825 

829 

832 

836 

839 

38 

53 

808 

812 

816 

819 

823 

826 

830 

834 

837 

840 

844 

37 

54 

814 

818 

821 

825 

828 

832 

835 

838 

842 

845 

848 

36 

55 

9.820 

9-823 

9.827 

9.830 

9.833 

9-837 

9.840 

9-843 

9.847 

9.850 

9-853 

35 

56 

825 

829 

832 

835 

839 

842 

845 

84g 

851 

854 

858 

34 

57 

831 

834 

837 

841 

844 

847 

850 

853 

856 

859 

862 

33 

58 

836 

840 

843 

846 

849 

852 

855 

858 

861 

864 

866 

32 

59 

842 

845 

848 

851 

854 

857 

860 

863 

865 

868 

871 

31 

114° 

115° 

116° 

117° 

118° 

119° 

120° 

121° 

122° 

123° 

124 

1 

JLat. 

Co-! 
Lat. 

Polar  Distance. 

TABLE  XXX. 


123 


Time-Azimuths:  Log  B. 

Declination. 

T  *•»*• 

Co- 

Jj£K». 

—24° 

—25° 

—26° 

—27° 

-28° 

—29° 

—30° 

—31° 

—32° 

—33° 

—34° 

Lat. 

0 

o 

0.672 

0.654 

0-637 

0.620 

0.603 

0.587 

0.572 

0-557 

0.542 

0.528 

0-515 

90 

1 

690 

671 

653 

635 

618 

60  1 

585 

57o 

555 

540 

526 

89 

2 

708 

688 

669 

651 

633 

616 

599 

583 

568 

552 

538 

88 

3 
4 

727 
747 

706 
725 

686 
704 

667 
684 

649 

665 

631 
646 

628 

597 
611 

581 
594 

578 

550 
563 

87 
86 

5 

0.768 

0-745 

0.723 

0.702 

0.682 

0.663 

0.644 

0.626 

0.609 

0.592 

0.576 

85 

5 

791 

766 

743 

721 

700 

680 

660 

642 

624 

606 

589 

84 

7 

814 

788 

764 

719 

698 

677 

658 

639 

621 

603 

83 

839 

812 

786 

762 

738 

716 

695 

675 

655 

636 

618 

82 

9 

866 

837 

810 

784 

759 

736 

692 

672 

652 

633 

81 

1O 

0.895 

0.864 

0-835 

0.807 

0.781 

0-757 

0-733 

0.711 

0.690 

0.669 

0.649 

8O 

11 

925 

892 

861 

832 

805 

779 

754 

730 

708 

687 

666 

79 

12 

959 

923 

890 

859 

829 

776 

727 

705 

683 

78 

13 

995 

956 

920 

887 

856 

827 

799 

773 

748 

724 

702 

77 

14 

1-035 

993. 

954 

918 

884 

853 

824 

796 

770 

745 

721 

76 

15 

1.080 

I-°33 

0.990 

0.951 

0.915 

0.881 

0.850 

0.820 

0-793 

0.766 

0.741 

75 

16 

129 

077 

1.030 

987 

948 

912 

878 

847 

817 

789 

763 

74 

17 

1  86 

127 

074 

1.027 

984 

945 

909 

875 

843 

814 

786 

73 

18 

25  l 

183 

124 

071 

1.024 

981 

941 

9°5 

871 

840 

810 

72 

19 

329 

248 

180 

1  20 

068 

1.020 

977 

938 

902 

868 

836 

71 

23 

1.424 

1.326 

1-245 

1.177 

1.117 

1.064 

1.017 

0.974 

0-934 

0.898 

0.864 

70 

21 

548 

421 

323 

242 

J73 

114 

06  1 

1.013 

970 

931 

894 

69 

22 

722 

544 

418 

319 

238 

170 

no 

057 

I.OIO 

966 

927 

68 

23 

2.022 

719 

414 

316 

235 

166 

1  06 

053 

i.  006 

962 

67 

24 

+  00 

2.018 

715 

538 

411 

3I2 

231 

162 

102 

049 

I.OOI 

66 

25 

2.018 

+  00 

2.015 

1.712 

1-534 

1.407 

1.308 

1.227 

I.I58 

1.098 

1.045 

65 

26 

I-7I5 

2.015 

+  00 

2.01  1 

708 

53° 

403 

3°4 

223 

154 

094 

.64 

27 

538 

1.712 

2.OII 

+  00 

2.007 

704 

526 

399 

300 

219 

150 

63 

2§ 

411 

534 

1.708 

2.007 

+  00 

2.003 

700 

522 

395 

296 

214 

62 

29 

312 

407 

530 

1.704 

2.003 

-h  °° 

999 

696 

390 

291 

61 

SO 

I.23I 

1.308 

1.403 

1.526 

1.700 

1.999 

+  00 

1.994 

1.691 

I-5I3 

1.386 

60 

31 

162 

227 

3°4 

399 

522 

696 

J-995 

+  00 

99° 

686 

508 

59 

33 

102 

158 

223 

300 

395 

5i7 

691 

1.990 

+  00 

985 

682 

58 

33 

049 

098 

154 

219 

296 

390 

513 

686 

1.985 

+  00 

980 

57 

34 

002 

°45 

094 

150 

214 

291 

386 

508 

682 

1.980 

+  00 

56 

35 

0.958 

0-997 

I.04I 

1.089 

I-I45 

I.2IO 

1.286 

1.381 

I-503 

1-677 

1-975 

55 

36 

954 

0-993 

036 

085 

140 

205 

281 

376 

'  498 

671 

54 

37 

88  1 

914 

949 

0.988 

031 

080 

135 

200 

276 

371 

493 

53 

33 

847 

877 

909 

944 

0.983 

O26 

°75 

130 

195 

271 

365 

52 

33 

815 

842 

872 

904 

939 

0.978 

021 

070 

125 

189 

265 

51 

4D 

0.785 

0.810 

0.838 

0.867 

0.899 

0-934 

0.973 

1.016 

1.064 

1.119 

1.183 

50 

41 

756 

780 

805 

833 

862 

894 

929 

0.968 

Oil 

059 

114 

49 

42 

729 

75  i 

775 

800 

828 

857 

889 

924 

0.962 

005 

053 

48 

!  43 
44 

703 
679 

724 
698 

746 
719 

770 

795 
764 

822 
789 

85I 

816 

846 

877 

0.956 
912 

0.999 
950 

47 
46 

45 

0.655 

0.674 

0.693 

0.714 

0-735 

0-759 

0.784 

0.811 

0.840 

0.871 

0.906 

45 

43 

47 

633 
611 

650 
627 

668 
645 

688 
663 

708 
682 

73° 
702 

753 
724 

778 
747 

805 
772 

8 

865 
827 

44 
43 

4§ 

590 

605 

622 

639 

657 

676 

696 

718 

765 

792 

42 

49 

570 

584 

600 

616 

633  ' 

651 

670 

690 

711 

734 

759 

41 

50 

0.564 

0-579 

0-594 

0.610 

0.627 

0.645 

0.663 

0.683 

0.705 

0.727 

40 

51 

531 

544 

558 

573 

588 

604 

620 

638 

657 

877 

698 

39 

52 

513 

525 

538 

552 

566 

581 

597 

614 

631 

650 

670 

38 

53 
54 

495 
477 

.507 
489 

519 

532 

546 
526 

560 
539 

575 
553 

590 
568 

607 
583 

624 
600 

643 
617 

37 
36 

55 

0.460 

0.471 

0.482 

0.494 

0.506 

0.519 

0-532 

0.546 

0.561 

0.576 

0-593 

35 

56 

444 

454 

465 

476 

487 

499 

512 

525 

539 

553 

569 

34 

57 

428 

437 

448 

458 

469 

480 

492 

505 

518 

532 

546 

33 

58 

412 

421 

43  i 

441 

462 

473 

485 

497 

5IQ 

524 

32 

59 

396 

4°5 

414 

424 

434 

444 

455 

466 

477 

490 

502 

31 

Lat. 

114° 

115° 

116° 

117° 

118° 

119° 

120° 

121° 

122° 

123° 

124° 

Co- 
Lat. 

Polar  Distance. 

124 


TABLE  XXX.  A. 


0* 

Time-  Azimuths  :  Log  C. 

m 

Oa.O 

0~.l 

0'n.2 

Oin.3 

Om.4 

Om.5 

Om.6 

On'.7 

Om.8 

Om.9 

lm.O 

m 

O 

-f  00 

3.661 

3-360 

3.184 

3-059 

2.962 

2.883 

2.816 

2.758 

2.707 

2.661 

59 

1 

2,66  1 

2.620 

2.582 

2-547 

2.515 

485 

457 

431 

406 

382 

360 

58 

2 

360 

339 

3i9 

299 

281 

263 

246 

230 

214 

199 

184 

57 

3 

184 

170 

156 

H3 

130 

117 

105 

093 

08  1 

070 

059 

56 

4 

059 

048 

038 

028 

018 

008 

1.998 

1.989 

1.980 

1.971 

1.962 

55 

5 
6 

1.962 
883 

1-954 
876 

!-945 
869 

1-937 
§62 

1.929 
855 

1.921 
848 

1.913 

842 

1.905 
835 

1.898 
829 

1.890 
822 

1.883 
816 

54 

53 

7 

816 

810 

804 

798 

792 

786 

780 

775 

769 

764 

758 

52 

8 
9 

758 
707 

753 
.702 

747 
697 

742 
693 

688 

732 
683 

727 
679 

722 
674 

717 
670 

712 

666 

707 
66  1 

51 

50 

10 

i.65i 

1-657 

1.653 

1.648 

1.644 

1.640 

1.636 

1.632 

1.628 

1.624 

1.620 

49 

11 

620 

616 

612 

608 

604 

600 

597 

593 

589 

586 

582 

48 

12 

582 

578 

575 

571 

568 

564 

561 

557 

554 

547 

47 

13 

547 

544 

541 

537 

534 

531 

528 

524 

r  ig 

46 

11 

5*5 

512 

509 

506 

503 

500 

497 

494 

491 

488 

485 

45 

15 

1.485 

1.482 

1-479 

1.476 

1-474 

1.471 

1.468 

1.465 

1.462 

1.460 

1-457 

44 

16 

457 

454 

452 

449 

446 

444 

441 

438 

436 

433 

43 

!  17 

43* 

428 

425 

423 

420 

418 

415 

413 

411 

408 

406 

42 

19 

4o5 

382 

403 

380 

401 
378 

399 
375 

396 
373 

394 

369 

389 
366 

387 
364 

384 
362 

382 
360 

41 
40 

,  2D 

1.360 

1-358 

1-356 

J-353 

1.351 

1-349 

1-347 

1-345 

1-343 

I-34I 

1-339 

39 

21 

339 

337 

335 

333 

330 

328 

326 

324 

322 

320 

38 

1  *j£ 

3!8 

316 

315 

313 

3" 

3°9 

307 

3°5 

303 

301 

299 

37 

23 

299 

297 

295 

293 

292 

290 

288 

286 

284 

282 

281 

36 

21 

281 

279 

277 

275 

273 

272 

270 

268 

266 

265 

263 

35 

2.5 

1.263 

1.261 

1.259 

1.258 

1.256 

1.254 

1-253 

1.251 

1.249 

1.247 

1.246 

34 

!  23 

246 

244 

242 

241 

239 

237 

236 

234 

233 

231 

229 

33 

27 

229 

228 

226 

225 

223 

221 

220 

218 

217 

215 

214 

32 

23 

214 

212 

2IO 

209 

207 

206 

204 

203 

20  1 

200 

198 

31 

23 

19$ 

197 

195 

194 

192 

191 

189 

188 

186 

I85 

183 

30 

,  33 

1.183 

I.I82 

1.181 

1.179 

1.178 

I.I76 

I.I75 

I-I73 

1.172 

I.I7I 

1.169 

29 

31 

169 

168 

166 

165 

164 

162 

161 

159 

158 

J55 

28  i 

1  3.2 

155 

154 

153 

151 

150 

149 

H7 

146 

H5 

H3 

142 

27 

33 

142 

141 

139 

138 

'37 

134 

133 

132 

130 

129 

26  ! 

34 

129 

128 

126 

125 

124 

I23 

121 

1  20 

119 

118 

116 

25 

35 

I.II6 

1.115 

1.114 

1.113 

i.  in 

I.IIO 

I.I09 

1.108 

1.106 

1.105 

1.104 

24 

33 

104 

103 

102 

100 

099 

098 

097 

096 

094 

093 

092 

23 

37 

092 

091 

090 

089 

087 

086 

085 

084 

083 

082 

080 

22 

38 

080 

079 

078 

077 

076 

°75 

074 

072 

071 

070 

069 

21  | 

33 

069 

068 

067 

066 

065 

064 

062 

061 

060 

059 

058 

20 

40 

1.058 

1.057 

1.056 

1-055 

1.054 

1-053 

1.052 

1.050 

1.049 

1.048 

1.047 

19 

41 

047 

046 

045 

044 

043 

042 

041 

040 

039 

038 

°37 

18 

:  42 

037 

036 

035 

°34 

°33 

032 

03I 

030 

029 

027 

026 

17 

43 

026 

025 

O24 

023 

022 

021 

020 

019 

018 

017 

016 

16  ; 

41 

016 

OI4 

013 

OI2 

on 

Oil 

OIO 

009 

008 

007 

15 

45 

1.007 

i.  006 

1.005 

1.004 

1.003 

i.  002 

I.OOI 

I.OOO 

0.999 

0.998 

0-997 

14 

:  43 

0.997 

0.996 

0-995 

0.994 

0-993 

0.992 

0.991 

0.990 

989 

989 

988 

13! 

47 

;  48 

988 
978 

987 
977 

986 

977 

985 
976 

984 

975 

983 
974 

982 

973 

981 

972 

980 
971 

979 
970 

978 
969 

12 
11 

49 

969 

968 

968 

957 

966 

965 

964 

963 

962 

961 

961 

10 

|  50 

0.961 

0.960 

0.959 

0.958 

0-957 

0.956 

0-955 

0.954 

0-954 

0-953 

0.952 

9 

i  51 

952 

95  * 

950 

949 

948 

948 

947 

946 

945 

944 

943 

8  ; 

j  52 

943 

942 

942 

941 

940 

939 

938 

937 

937 

936 

935 

7 

53 

935 

934 

933 

933 

932 

93° 

929 

928 

928 

927 

6  ! 

54 

927 

926 

925 

924 

924 

923 

922 

921 

920 

920 

919 

5 

55 

0.919 

0.918 

0.917 

0.916 

0.916 

0.915 

0.914 

0.913 

0.912 

0.912 

0.911 

4 

56 

911 

910 

909 

909 

908 

907 

906 

905 

905 

904 

9°3 

3  i 

57 

903 

902 

902 

901 

900 

899 

899 

898 

895 

2 

i  58 

895 

895 

894 

893 

892 

892 

890 

889 

888 

888 

1 

59 

888 

887 

886 

886 

885 

884 

884 

883 

882 

88  1 

881 

I 

m 

1-0 

0-.9 

Om.8 

Om.7 

Om.6 

Om.5 

Om.4 

0-3 

Om.2 

Om.l 

Om.O 

m 

With  Hour-  Angles  greater  than  6h  read  from  bottom,  and  subtract  tabular  [  1  lh 

Log  from  10.000. 

TABLE  XXX.  A. 


125 


lh 

Time-Azimuths:   Log  €. 

m 

O".O 

Om.l 

Om.2 

On.3 

0'».4 

O;n.5 

Ora.6 

0:n.7 

O!n.8 

Om.9 

1-0 

'm   ! 

O 

0.88  1 

0.880 

0.879 

0.878 

0.878 

0.877 

0.876 

0.875 

0.875 

0.874 

0-873 

59  ! 

i 

873 

873 

872 

871 

870 

870 

869 

868 

868 

857 

855 

58  ! 

2 

866 

865 

865 

864 

863 

853 

862 

861 

86  1 

860 

859 

57 

3 

859 

858 

858 

857 

856 

856 

855 

854 

854 

853 

852 

53  : 

4 

852 

852 

851 

850 

849 

849 

848 

847 

847 

846 

845 

55 

5 

0.845 

0.845 

0.844 

0.843 

0.843 

0.842 

0.841 

0.841 

0.840 

0.839 

0.839 

>  i  ! 

6 

839 

838 

837 

837 

836 

835 

835 

834 

833 

833 

832 

53  i 

7 

832 

831 

831 

830 

829 

829 

828 

827 

827 

826 

826 

52 

8 

826 

825 

824 

824 

823 

822 

822 

821 

820 

820 

819 

51  i 

9 

819 

818 

818 

817 

817 

816 

815 

815 

814 

813 

813 

50 

1O 

0.813 

0.812 

0.811 

0.811 

0.810 

0.810 

0.809 

0.808 

0.808 

0.807 

0.806 

49  ! 

11 

806 

806 

805 

805 

804 

803 

803 

802 

802 

831 

800 

48  i 

12 

800 

800 

799 

798 

798 

797 

797 

796 

795 

795 

794 

47  : 

794 

794 

793 

792 

792 

791 

791 

790 

789 

789 

788 

46 

14 

788 

783 

787 

786 

786 

785 

785 

784 

783 

783 

782 

45  j 

15 

0.782 

0.782 

0.781 

0.780 

0.780 

0.779 

0.779 

0.778 

0.778 

0.777 

0.776 

44  i 

16 

776 

775 

775 

775 

774 

773 

773 

772 

772 

771 

771 

43  ! 

771 

770 

769 

769 

758 

768 

767 

767 

766 

765 

765 

42 

1    £8 

7^5 

764 

764 

763 

753 

762 

762 

761 

760 

700 

759 

41  j 

19 

759 

759 

758 

758 

757 

756 

756 

755 

755 

754 

754 

40 

2O 

0-754 

0-753 

0-753 

0.752 

0.751 

0.751 

0.750 

0.750 

0.749 

0-749 

0.748 

39 

21 

748 

748 

747 

747 

746 

745 

745 

744 

744 

743 

743 

38  i 

22 

743 

742 

742 

741 

740 

739 

739 

738 

738 

737 

37 

23 

737 

737 

736 

736 

735 

735 

734 

734 

733 

733 

732 

36 

24 

732 

732 

73  i 

730 

730 

729 

729 

728 

728 

727 

727 

35 

25 

0.727 

0.726 

0.726 

0.725 

0.725 

0.724 

0.724 

0.723 

0.723 

0.722 

0.722 

34 

26 

722 

721 

721 

720 

720 

719 

718 

718 

717 

717 

716 

33 

27 

716 

716 

715 

7H 

714 

713 

713 

712 

712 

711 

32 

:  2§ 

711 

711 

710 

710 

709 

709 

708 

7oS 

707 

707 

706 

31  ! 

29 

706 

7o5 

705 

705 

7^4 

704 

703 

703 

702 

702 

701 

30 

30 

0.701 

0.701 

0.700 

0.700 

0.699 

0.699 

0.698 

0.698 

0.697 

0.697 

0.696 

29 

31 

696 

696 

695 

695 

694 

694 

693 

693 

692 

28  | 

!  32 

692 

691 

690 

690 

689 

689 

688 

688 

687 

687 

27 

33 

687 

635 

685 

685 

685 

684 

684 

683 

683 

682 

682 

26 

34 

682 

68  1 

68  1 

6Si 

680 

680 

679 

679 

678 

678 

677 

25 

35 

0.677 

0.677 

0.676 

0.676 

0.675 

0.675 

0.674 

0.6  "4 

0.673 

0.673 

0.673 

24 

36 

673 

672 

672 

671 

671 

670 

670 

669 

669 

668 

668 

23 

37 

668 

667 

667 

657 

666 

666 

665 

665 

664 

664 

663 

22 

38 

663 

663 

662 

662 

661 

661 

661 

660 

660 

659 

659 

21 

39 

659 

658 

658 

657 

657 

656 

656 

656 

655 

655 

654 

20 

40 

0.654 

0.654 

0-653 

0.653 

0.652 

0.652 

0.652 

0.651 

0.651 

0.650 

0.650 

19 

41 

650 

649 

649 

648 

•    648 

648 

647 

647 

646 

646 

645 

18 

42 

645 

645 

644 

644 

644 

643 

643 

642 

642 

641 

641 

17 

43 

641 

641 

640 

640 

639 

639 

638 

638 

638 

637 

637 

16  ! 

44 

637 

636 

630 

635 

635 

634 

634 

634 

633 

633 

632 

15 

45 

0.632 

0.632 

0.631 

0.631 

0.631 

0.630 

0.630 

0.629 

0.629 

0.628 

0.628 

14  ; 

!  46 

628 

628 

627 

627 

626 

626 

626 

625 

625 

624 

624 

13 

!  47 

624 

623 

623  1      623 

622 

622 

621 

621 

620 

620 

620 

12 

!  48 

620 

619 

619  i       618 

618 

618 

617 

617 

616 

616 

615 

11 

49 

615 

6i5 

615         614 

614 

613 

613 

613 

612 

612 

611 

1O 

!  5*> 

0.611 

0.611 

0.611 

0.610 

0.6  10 

0.609 

0.609 

0.609 

0.608 

0.608 

0.607 

9 

51 

607 

607 

606 

606 

606 

605 

605 

604 

604 

604 

603 

8 

52 

603 

603 

602 

602 

602 

60  1 

60  1 

600 

600 

600 

599 

7 

53 

599 

599 

598 

598 

598 

597 

597 

596 

596 

596 

595 

6 

54 

595 

595 

594 

594 

594 

593 

593 

592 

592 

592 

591 

5 

J55 

0.591 

0.591 

0.590 

0.590 

0.590 

0.589 

0.589 

0.589 

0.588 

0.587 

0.587 

4 

56 

587 

587 

587 

586 

586 

585 

585 

585 

584 

584 

583 

3 

57 

583 

583 

583 

582 

582 

582 

581 

,581 

580 

580 

580 

2 

58 

580 

579 

579 

578 

578 

578 

577 

577 

577 

576 

5/6 

1 

59 

576 

575 

575 

575 

574 

574 

573 

573 

573 

572 

572 

0 

m 

1».0 

0-.9 

Om.8 

0^.7 

O:n.6 

0-.5 

Om.4 

Om.3 

Om.2 

Om.l 

Oin.O 

m 

"With  Hour-  Angles  greater  than  6b  read  from  bottom,  and  subtract  tabular    L  *  **  l 

Log  from  14X000. 

126 


TABLE  XXX.  A. 


9h 

Time-Azimuths:  Log:  C. 

t 

m 

o».o 

O-.l 

0'u.2 

0".3 

0«,4 

0».5 

0».6 

Om.7 

0-8 

0™9 

1>0 

m 

0 

0.572 

0.572   0.571 

0571 

0.570 

0.570 

0.570 

0.569 

0.569 

0.569 

0.568 

59 

1 

558 

568    567 

567 

567 

566 

566 

566 

565 

565    5$4 

58 

2 

504 

564    564 

5^3 

5^3 

563 

562 

562 

561 

561    561 

57 

3 

55o    560 

560 

559 

559 

558 

558 

558  j   557    557 

56 

4 

557 

557  !   55$ 

55$ 

556 

555 

555 

554 

554 

554 

553 

55 

5 

o-553 

o-553   0.553 

0.552 

0-552 

0.552 

0.551 

0.551 

0.550 

0-55° 

0.550 

54 

6 

550 

549    549 

549 

548 

548 

548 

547 

547 

54$    54$ 

53 

7 

54$ 

54^    545 

545 

545 

544 

544 

544 

543 

543    543 

52 

8 

543 

542    542 

542 

541 

541 

540 

540 

540 

539    539 

51 

9 

539 

539  ;  538 

538 

538 

537 

537 

536 

53$ 

53$    535 

5O 

19 

0-535 

o-535   o-535 

0-534 

0-534 

0-534 

o-533 

0-533 

0.533 

0.532  |  0.532 

49 

11 

532 

532    53' 

531 

530 

530 

530 

529 

529 

529  ;   528 

48 

Id 

528 

528    528 

527 

527 

527 

526 

526 

525 

525    525 

47 

13 

525 

525  ;  524 

524 

524 

523 

523 

523 

522 

522  :   521 

48 

14 

521 

521   521 

520 

520 

520 

5^9 

5'9 

519 

45 

15 

0.518 

0.518  0.517 

0.517 

0.517 

0.516 

0.516 

0.516 

0.515 

0.515  i  0.515 

44 

1  16 

515  i   514    514 

514 

513 

513 

513 

512 

512 

512    511 

43 

17 

5"    5"    511 

5*° 

5ro 

510 

5°9 

5°9 

5°9 

508  i   508 

42 

13 

508 

508  .  5^7 

507    507 

506 

506 

506 

505 

505  :   505 

41 

19 

505 

504   504   504 

533 

503 

503 

502  j   502 

502  \   501 

40 

1  2D 

0.501 

0.501  i  0.501  0.500 

0.500 

0.500 

0-499 

0.499   0.499 

0.498  !  0.498 

39 

21 

498 

498  '   497    497 

497 

496 

496 

49$    495 

495    495 

38 

2«J 

495 

494    494 

494 

493 

493 

493 

492    492 

492  !    49» 

37 

23 

491 

491  i   491 

493 

493 

490 

493 

489    489 

489    483 

36 

24 

488 

488  l   488 

487 

487 

487 

486 

486 

.  485 

485    485 

35 

25 

0.485 

0485   0.484 

0.484 

0.484 

0.483 

0.483 

0.483 

0.482 

0.482  1  0.482 

34 

23 

482 

481 

481 

481 

481 

480 

480 

480 

479 

479    479 

33 

27 

479 

478 

478 

478 

477 

477 

477 

476 

47$ 

47$    475 

32 

2§ 

475 

475 

475 

475 

474 

474 

474 

473 

4/3 

473 

472 

31 

29 

472 

472 

472 

47i 

471 

47° 

470 

470 

470 

469 

30 

30 

0.469 

0.469 

0.469 

0.468 

0.468 

0.468 

0.467 

0.467 

0.467 

0.466 

0.466 

29 

31 

'  466 

466 

4<>5 

465 

465 

465 

464 

464 

464 

4$3 

4$3 

28 

32 

463 

463 

462 

462 

462 

461 

461 

461 

461 

460 

460 

27 

33 

460 

460 

459 

459 

459 

458 

458 

458 

458 

457 

457 

26 

34 

457 

457 

45$ 

45  $ 

456 

455 

455 

455 

454 

454 

454 

25 

35 

0-454 

0-454 

0-453 

0-453 

0-453 

0.452 

0.452 

0.452 

0-45  l 

0.451 

0.451 

24 

36 

451 

45° 

45° 

45° 

449 

449 

449 

448 

448 

448 

23 

37 

448 

448 

447 

447 

447 

446 

446 

446 

445 

445 

445 

22 

33 

445 

445 

444 

444 

444 

443 

443 

443 

442 

442 

442 

21 

39 

442 

442 

441 

441 

441 

44° 

440 

440 

440 

439 

439 

2O 

40 

o-439 

0-439 

0.438 

0.438 

0.438 

0-437 

0-437 

0.437 

0-437 

0,436 

0.436 

19 

41 

436 

43$ 

435 

435 

435 

435 

434 

434 

434 

433 

433 

18  ! 

42 

433 

433  I   432 

432 

432 

432 

43  l 

43° 

43° 

17 

43 

43° 

43°  1   430 

429 

429 

429 

4~28 

428 

428 

428 

427 

16 

41 

427 

427    427 

426 

426 

426 

426 

425 

425 

425 

424 

15 

45 

0.424 

0.424 

0.424 

0.424 

0.423 

0.423 

0.423 

0.422 

0.422 

0.422 

0.422 

14 

46 

422 

421 

421 

421 

420 

420 

420 

420 

419 

419 

419 

13 

47 

419 

418 

418 

418 

418 

417 

417 

417 

416 

416 

416 

12 

48 

416 

416 

415 

415 

415 

414 

414 

414 

414 

413 

413 

11 

49 

413 

413 

412 

412 

412 

412 

411 

411 

411 

410 

410 

1O 

50 

0.410 

0.410 

0.410 

0.409 

0.409 

0.409 

0.409 

0.408 

0.408 

0.408 

0.407 

9 

51 

407 

407 

407 

406 

406 

406 

405 

4°5 

405 

405 

8 

52 

405 

404 

404 

404 

403 

403 

403 

403 

402 

402 

402 

7 

53 

402 

402 

401 

401 

401 

400 

400 

400 

400 

399 

399 

6 

54 

399 

399 

399 

398 

398 

398 

397 

397 

397 

397 

39$ 

5 

55 

0.396 

0.395 

0.396 

0.396 

0-395 

o-395 

o.395 

0-394 

0-394 

0-394 

0-394 

i 

56 

394 

393 

393 

393 

393 

392 

392 

392 

39i 

391 

3io 

3 

57 

39* 

39i 

39° 

393 

390 

390 

389 

389 

389 

388 

388 

i  > 

58 

388 

388 

388 

387 

387 

387 

387 

385 

386 

386 

385 

1 

59 

385 

385 

385 

385 

384 

384 

384 

384 

383 

383 

383 

0 

m 

1-0 

On.9 

Om.8 

Om.7 

Ora.6 

0-5 

Om.4 

Om.3 

Om.2 

Om.l 

O'".O 

m 

With  Hour-  Angles  greater  than  6h  read  from  bottom,  and  subtract  tabular 

9h 

Log  from  10.000. 

TABLE  XXX.  A. 


127 


3h 

Time-  Azimuths:  Log  C. 

m 

0'".0 

O'M 

Om.2 

0™.3 

Om.4 

Om.5 

Om.6 

0™.7 

0™.8 

0™.9 

lm.O 

m 

O 

0-383 

0-383 

0.382 

0.382 

0.382 

0.381 

0.381 

0.381 

0.381 

0.380 

0.380 

59 

1 

380 

380 

380 

379 

379 

379 

379 

378 

378 

378 

377 

58 

» 

377    377 

377 

377 

376 

376 

376 

376 

375 

375 

375 

57 

3 

375    375 

374 

374 

374 

373 

373 

373 

373 

372 

372 

56 

* 

372    372 

372 

37i 

37i 

37i 

37i 

370 

37o 

37» 

370 

55 

5 

0.370  !  0.369 

0.369 

0.369 

0.368 

0.368 

0.368 

0.368 

0.367 

0.367 

0.367 

54 

6 

367    367 

366 

366 

366 

366 

365 

365 

365 

365 

364 

53 

f 

364  |   364 

364 

364 

363 

363 

363 

362 

362 

362 

362 

52 

§ 

362  ;   361 

361 

361 

361 

360 

360 

360 

360 

359 

359 

51 

9 

359  |   359 

359 

358 

358 

358 

358 

357 

357 

357 

357 

50 

10 

0-357  i  0.356 

0-356 

0-356 

0-356 

0-355 

0-355 

0-355 

0-354 

0-354 

o-354 

49 

11 

354  !   354 

353 

353 

353 

353 

352 

352 

352 

352 

351 

48 

12 

35i    351 

35i 

35i 

350 

350 

350 

350 

349 

349 

349 

47 

13 

349    349 

348 

348 

348 

348 

347 

347 

347 

347 

346 

46 

14 

346  |   346 

346 

346 

345 

345 

345 

345 

344 

344 

344 

45 

15 

0.344   0.344 

o-343 

o-343 

o-343 

o-343 

0.342 

0.342 

0.342 

0.342 

0.341 

44 

16 

34i    34i 

34i 

34i 

340 

340 

340 

340 

339 

339 

339 

43  ! 

17 

339    339 

33f 

338 

338 

338 

337 

337 

337 

337 

336 

42  ; 

IS 

336  :   336 

336 

336 

335 

335 

335 

335    334 

334 

334 

41 

19 

334  !   334 

333 

333 

333 

333 

332 

332 

332 

332 

33i 

40  i 

20 

0-331   0.331 

o-33i 

0-331 

0-330 

0-330 

0-33° 

0-33° 

0.329 

0.329  . 

0.329 

39  j 

91 

329  1   329    328 

328 

328 

328 

327 

327 

327 

327 

326 

38  1 

1  22 

326  !   326    326 

326 

3*5 

325 

325 

325 

324 

324 

324 

37 

23 

324    324    323    323 

323 

323 

322 

322 

322 

322 

322 

36 

24 

322  ;   321    321 

321 

321 

320 

320 

320 

320 

319 

3*9 

35  i 

25 

0.319 

0.319 

0.319 

0.318 

0.318 

0.318 

0.318 

0.317 

0.317 

o-3i7 

0-317 

34 

26 

317 

316 

316 

316 

316 

315 

315 

315 

315 

3H 

3*4 

33 

27 

3H 

3H 

3H 

3H 

3i3 

3J3 

313 

313 

312 

312 

312 

32  1 

28 

312 

312 

3ii 

3" 

3" 

3H 

310 

310 

310 

310 

3°9 

31 

29 

309 

3°9 

3°9 

309 

308 

308 

308 

308 

308 

307 

307 

3O 

3O 

0-30.7 

0.307 

0.307 

0.306 

0.306 

0.306 

0.306 

0-305 

0.305 

0-305 

0-305 

29 

31 

305 

304 

304 

3°4 

3°4 

3°3 

303 

303 

303 

303 

302 

•28  i 

32 

302 

302 

3°2 

302 

301 

301 

301 

301 

300 

300 

3°° 

27 

33 

300 

300 

299 

299 

299 

299 

298 

298 

298 

298 

298 

26 

34 

298 

297 

297 

297 

297 

296 

296 

296 

296 

295 

295 

25 

35 

0.295 

0.295 

0.295 

0.294 

0.294 

0.294 

0.294 

0.294 

0.293 

0.293 

0.293 

24 

36 

293 

293 

292 

292 

292 

292 

291 

291 

291 

290 

23 

37 

290 

290 

290 

290 

290 

289 

289 

289 

289 

288 

288 

22 

38 

288    288 

288 

287 

287 

•  287 

287 

287 

286 

286 

286 

21 

39 

286 

286 

285 

285 

285 

285 

284 

284 

284 

284 

284 

20 

4O 

0.284 

0.283 

0.283 

0.283 

0.283 

0.282 

0.282 

0.282 

0.282 

0.281 

0.281 

19 

41 

281 

281 

281 

281 

286 

280 

280 

280 

279 

279 

279 

18 

42 

279 

279 

278 

278 

278 

278 

278 

277 

277 

277 

277 

17 

43 

277 

276 

276 

276 

276 

275 

275 

275 

275 

275 

274 

16 

44 

274 

274 

274 

274 

273 

273 

273 

273 

272 

272 

272 

15 

45 

0.272 

0.272 

0.272 

0.271 

0.271 

0.271 

0.271 

0.270 

0.270 

0.270 

0.270 

14 

46 

270 

270 

269 

269 

269 

269 

268 

268 

268 

268 

267 

13 

47 

267 

267 

267 

267 

267 

266 

266 

266 

266 

265 

265 

12 

48 
49 

265 
263 

265 
263 

265 
263 

265 
262 

264 
262 

264 
262 

264 
262, 

264 
261 

263 
261 

4 

261 

4 

261 

11 
10 

50 
51 

0.261 
258 

0.261 

258 

0.260 
258 

0.260 
258 

0.260 
258 

0.260 

257 

0.259 
257 

0.259 
257 

0.259 

257 

0.259 
256 

0.258 
256 

9 

8 

52 

256 

256 

256 

256 

255 

255 

255 

255 

254 

254 

254 

7 

53 

254 

254 

254 

253 

253 

253 

253 

252 

252 

252 

252 

6 

54 

252 

252 

251 

25  i 

251 

251 

250 

250 

250 

250 

250 

5 

55 

0.250 

0.249 

0.249 

0.249 

0.249 

0.248 

0.248 

0.248 

0.248 

0.248 

0.247 

4 

56 

247 

247 

247 

247 

246 

246 

246 

246 

246 

245 

245 

3 

57 

245 

245 

245 

244 

244 

244 

244 

244 

243 

243 

243 

2 

58 

243 

243 

243 

242 

242 

242 

242 

241 

241 

241 

241 

1 

59 

241 

241 

240 

240 

240 

240 

239 

239 

239 

.239 

239 

0 

m 

lm.O 

Om.9 

Ora.8 

Ora-7 

Om.6 

Om.5 

Om.4 

Om.3 

Om.2 

Ora.l 

Om.O 

m 

With  Hour-  Angles  greater  than  611  read  from  bottom,  and  subtract  tabular 

8h| 

Log  from  10.000. 

128 


TABLE  XXX.  A. 


4'.] 

Time-Azimuths:  Log  €. 

m 

0'".0 

0-1 

0".2 

0  .3 

O:u  4 

0».5 

O".6  On.7  O!1'.8 

O:".9  ;  1"'.O 

111 

0 

0.239 

0.238 

0.238 

0.238 

0.238 

0.237 

0.237  !  0.237   0.237 

o.  37   0.236 

59 

1 

236 

236 

236 

236 

236 

235 

235  !   235    235     ^4    234 

5S 

2 

234 

234 

234 

-.U    233 

233 

233    233  !   232  j    ^2    232 

57 

3 

232 

232 

232 

231  i   231 

231 

231 

231  ;   230  \    30     230 

.53 

4 

230 

230 

229 

229 

229 

229 

229 

228  j   228  j    .8  i   228 

5.5 

5 

0.228 

0.227 

0.227 

0.227 

0.227 

0.227 

0.226 

0.226 

0.226 

0.226   0.226 

51 

6 

226 

225 

225 

225 

225 

224 

224 

224 

224 

224 

223 

53 

7 

223 

223 

223 

223 

223 

222 

222 

222 

222 

222 

222 

52 

8 

222 

221 

221 

221 

220 

220 

22O 

220 

22O 

219 

210 

51 

9 

219 

2I9 

2I9 

218 

218 

218 

218 

218 

2I7 

217     217 

50 

10 

0.217 

O.2I7 

0.217 

0.2  1  6 

0.216 

0.2  1  6 

0.216 

0.215 

O.2I5 

O.2I5    O.2I5 

49 

11 

215 

215 

2I4 

214 

214 

214 

214   213 

213 

2I3  ;    2I3 

48 

12 

2I3 

212 

212 

212 

212 

212 

211 

211 

211 

211      211 

47 

13 

211 

210 

2IO 

2IO 

2IO 

209 

209 

209 

209 

209  ;  208 

46 

14 

208 

208 

208 

208 

208 

207 

207 

207 

207 

207   206 

45 

15 

O.2O6 

0.206 

O.2O6 

0.206 

O.2O5 

0.205 

O.2O5 

O.2O5 

O.2O5 

O.2O4  !   0.2O4 

44 

16 

204 

204 

204 

204 

203 

203 

203 

203 

203 

202      202 

43 

17 

202 

202 

202 

2O  I 

201 

2OI 

201 

201 

200 

200     200 

42 

18 

200 

2OO 

2OO 

199 

199 

199 

199 

I99 

I98 

198  •    198 

11 

19 

I98 

198 

I97 

197 

197 

197 

197 

I96 

I96 

196     196 

40 

20 

0.195 

O.I95 

0.195 

0.195 

0.195 

0.195 

0.195 

0.194 

0.194 

0.194    0.194 

39 

21 

194 

I94 

193 

193 

193 

192 

192 

192 

192      192 

38 

22 

I92 

191 

191 

191 

191 

191 

190 

190 

I90 

190  '    190 

37 

23 

189 

189 

I89 

l89 

189 

1  88 

1  88 

1  88 

1  88 

I87 

36 

<  24 

187 

I87 

I87 

I87 

l87 

1  86 

1  86 

1  86 

1  86 

1  86 

I85 

35 

25 

0.185 

0.185 

0.185 

0.185 

0.185 

0.184 

0.184 

0.184 

0.184 

0.184 

0.183 

34 

26 

183 

!&3 

I83 

183 

183 

182 

182 

182 

182 

181 

181 

33 

27 

181 

181 

181 

181 

180 

1  80 

1  80 

1  80 

180 

179 

179 

32 

!  28 

17J 

179 

179 

179 

178 

178 

178 

178 

178 

177 

177 

31 

29 

177 

177 

177 

177 

176 

176 

176 

176 

176 

175 

'75 

30 

30. 

0.175 

0.175 

0.175 

o.i74 

0.174 

0.174 

0.174 

0.174 

0-173 

0,173 

°-I73 

29 

31 

173 

173 

'73 

172 

172 

172 

172 

172 

171 

171 

171 

28 

;  32 

171 

171 

170 

170 

170 

170 

170 

169 

169 

169 

27 

33 

169 

^69 

169 

1  68 

1  68 

1  68 

1  68 

168 

167 

167 

167 

26 

34 

167 

167 

167 

1  66 

1  66 

166 

166 

1  66 

165 

165 

165 

2,5 

35 

0.165 

0.165 

0.164 

0.164 

0.164 

0.164 

0.164 

0.163 

0.163 

0.163 

0.163 

24 

36 

163 

163 

162 

162 

162 

162 

162 

161 

161 

161 

161 

23 

37 

161 

161 

1  60 

1  60 

1  60 

1  60 

1  60 

!59 

'59 

159 

159 

22 

38 

'59 

159 

158 

158 

158 

158 

158 

'57 

'57 

157 

21 

39 

i57 

'57 

156 

156 

156 

156 

156 

155 

'55 

'55 

20 

40 

0-155 

0.155 

0.154 

0.154 

0.154 

0.154 

0.154 

0-153 

o-!53 

0-153 

0-153 

19 

41 

153 

153 

J52 

'52 

152 

152 

152 

151 

151 

151 

151 

18 

42 

151 

'51 

'5° 

IS° 

150 

149 

149 

149 

149 

17 

43 

149 

149 

148 

[48 

148 

148 

148 

H7 

H7 

147 

16 

44 

H7 

'47 

146 

146 

146 

146 

146 

*45 

'45 

'45 

'45 

15 

45 

0.145 

0.145 

0.144 

0.144 

0.144 

0.144 

0.144 

0.143 

0.143 

0.143 

0.143 

14 

46 

143 

143 

142 

142 

142 

142 

142 

141 

141 

141 

141 

13 

47 

141 

141 

140 

140 

140 

140 

140 

*39 

139 

i39 

139. 

12 

48 

139 

139 

'38 

138 

138 

138 

138 

»37 

137 

137 

137 

11 

49 

137 

136 

136. 

136 

136 

136 

135 

135 

1O 

50 

0.135 

o.i35 

0.134 

0.134 

0.134 

0.134 

0.134 

0.133 

0*133 

0.133 

0.133 

9 

51 

133 

i33 

132 

132 

132 

132 

132 

131 

131 

I3I 

131 

§ 

52 

131 

130 

130 

130 

130 

130 

129 

129 

129 

129 

7 

53 

129 

129 

128 

128 

128 

128 

128 

127 

127 

127 

12J 

(i 

54 

127 

127 

126 

126 

126 

126 

126 

I25 

125 

125 

125 

5 

55 

0.125 

0.125 

0.124 

0.124 

0.124 

0.124 

0.124 

0.123 

0.123 

0.123 

0.123 

1 

56 

123 

123 

122 

122 

122 

122 

122 

122 

121 

121 

121 

3 

57 

121 

121 

121 

I2O 

1  2O 

1  2O 

1  2O 

1  2O 

119 

119 

1  19 

9 

58 

119 

119 

119 

118 

118 

118 

118 

118 

117 

117 

117 

1 

59 

117 

117 

116 

116 

116 

116 

116 

"5 

H5 

0 

m 

l'r-.O 

Om.9 

0'".8 

Om.7 

0"'.6 

Om.5 

Om.4 

O'".3 

Om.2 

Om.l 

Oin.O 

m 

With  Hour-  Angles  greater  than  6h  read  from  bottom,  and  subtract  tabular 

r 

Log  from  10.000. 

TABLE  XXX.  A. 


129 


5h 

Time-  Azimuths:  Log  C. 

m 

Om.O 

Om.l 

O'".2 

0'».3  0»'.4 

0'».5 

0'».6 

0'".7 

0»\8 

0'».9 

1"'.0 

m  ; 

O 

0.115 

0.115 

0.115   0.114 

0.114 

0.114 

0.114 

0.114 

0.113 

0.113 

0.113 

59 

1 

113 

113 

113     112     112 

112 

112 

112 

in 

in 

III 

58 

!  * 

in 

III 

III     III      110 

IIO 

IIO 

IIO 

IIO 

109 

109 

57 

3 

109 

109 

109    109    108 

1  08 

1  08 

108 

108 

107    107 

56 

4 

107 

107    107 

107    106 

1  06 

106 

1  06 

1  06 

105    105 

55 

5 

0.105 

0.105   0.105 

0.105 

0.104 

O.IO4 

0.104 

0.104 

0.104 

0.103 

0.103 

54 

6 

103 

103 

103 

103 

103 

102 

102 

102 

1  02 

1  02 

IOI 

53 

7 

101 

IOI 

IOI 

IOI 

IOI 

IOO 

IOO 

IOO 

IOO 

IOO 

099 

52 

§ 

099 

099 

099 

099 

099 

098 

098 

098 

098 

098 

097 

51 

9 

097 

097 

097 

097 

097 

096 

096 

096 

096 

096 

096 

50 

1O 

0.096 

0.095 

0.095 

0.095 

0.095 

0.095 

0.094 

0.094 

0.094 

0.094 

0.094 

49  ! 

11 

094 

093 

°93 

093 

°93 

093 

092 

092 

092 

092 

092 

48 

13 

092 

091 

091 

091 

091 

091 

090 

090 

090 

090 

090 

47 

13 

090 

090 

089 

089 

089 

089 

089 

088 

088 

088 

088 

46 

14 

088 

088 

087 

087 

087 

087 

087 

086 

086 

086 

086 

45  i 

15 

0.086 

0.086 

0.085 

0.085 

0.085 

0.085 

0.085 

0.084 

0.084 

0.084 

0.084 

44 

16 

084 

084 

084 

083 

083 

083 

083 

083 

082 

082 

082 

43 

17 

082 

082 

082 

081 

08  1 

081 

081 

08  1 

080 

080 

080 

42 

18 

080 

080 

'  080 

079 

079 

079 

079 

079 

078 

078 

078 

41 

19 

078 

078 

078 

078 

077 

077 

077 

077 

077 

076 

076 

40 

20 

0.076 

0.076 

0.076 

0.076 

0.075 

'  0.075 

0.075 

0.075 

0.075 

0.074 

0.074 

39 

21 

074 

074 

074 

074 

073 

°73 

073 

073 

073 

073 

072 

38 

22 

072 

072 

072 

072 

072 

071 

071 

071 

071 

071 

070 

37 

23 

070 

070 

070 

070 

070 

069 

069 

069 

069 

069 

069 

36 

24 

069 

068 

068 

068 

068 

068 

067 

067 

067 

067 

067 

35 

25 

0.067 

0.066 

0.066 

0.066 

0.066 

0.066 

0.065 

0.065 

0.065 

0.065 

0.065 

34 

26 

065 

064 

064    064 

064 

064 

064 

063 

063 

063 

063 

33 

27 

063 

063 

062 

062 

062 

062 

062 

061 

06  1 

061 

061 

32 

|  2§ 

061 

06  1 

060 

060 

060 

060 

060 

059 

059 

°59 

059 

31 

29 

059 

059 

°59 

058 

058 

058 

058 

058 

057 

057 

057 

3O 

30 

0.057 

0.057 

0.057 

0.056 

0.056 

0.056 

0.056 

0.056 

o-055 

0.055 

0.055 

29 

31 

055 

055 

°55 

055 

054 

054 

054 

054 

054 

°53 

053 

28 

32 

053 

053 

°53 

053 

052 

052 

052 

052 

052 

051 

OS1 

27 

33 

051 

°5  J 

°5  ! 

OS1 

OS1 

050 

050 

05,0 

050 

050 

049 

26 

34 

049 

049 

049 

049 

049 

048 

048 

048 

048 

048 

047 

25 

35 

0.047 

0.047 

0.047 

0.047 

0.047 

0.047 

0.046 

0.046 

0.046 

0.046 

0.046 

24 

36 

046 

045 

045 

045 

045 

045 

044 

044 

044 

044 

044 

23 

37 

044 

043 

043 

043 

043 

043 

043 

042 

042 

042 

042 

22 

as 

042 

042 

041 

041 

041 

041 

041 

040 

040 

040 

040 

21 

39 

040 

040 

039 

039 

039 

°39 

039 

039 

038 

038 

038 

2O 

40 

0.038 

0.038 

0.038 

0.037 

0.037 

0.037 

0.037 

0.037 

0.036 

0.036 

0.036 

19 

41 

036 

036 

036 

035 

035 

035 

035 

035 

035 

034 

034 

18  i 

42 

034 

034 

034 

034 

°33 

033 

033 

033 

033 

032 

032 

17 

43 

032 

032 

032 

032 

031 

031 

031 

031 

031 

031 

030 

16 

44 

030 

030 

030 

030 

030 

029 

029 

029 

029 

029 

028 

15 

45 

0.028 

0.028 

0.028 

0.028 

0.028 

0.027 

0.027 

0.027 

0.027 

0.027 

0.027 

14  I 

46 

027 

026 

026 

026 

026 

026 

025 

025 

025 

025 

025 

13  1 

47 

025 

024 

024 

024 

024 

024 

024 

023 

023 

023 

023 

12 

48 

023 

023 

022 

022 

022 

022 

022 

021 

021 

O2I' 

021 

11 

49 

02  1 

021 

020 

O2O 

020 

O2O 

O2O 

O2O 

OI9 

OI9 

OI9 

1O  1 

50 

0.019 

O.OI9 

O.OI9 

o.o  1  8 

0.018 

0.018 

0.018* 

O.OlS 

0.017 

0.017 

0.017 

9 

51 

017 

017 

017 

016 

016 

016 

016 

016 

016 

015 

015 

8  I 

52 

°i5 

OI5 

°I5 

oi5 

014 

014 

014 

014 

014 

OI3 

OI3 

7  | 

53 

013 

013 

OI3 

013 

013 

OI2 

OI2 

012 

012 

012 

Oil 

6  j 

54 

•  on 

on 

on 

on 

Oil 

010 

OIO 

OIO 

OIO 

OIO 

009 

5 

55 

0.009 

0.009 

0.009 

0.009 

0.009 

O.OO9 

O.OO8 

0.008 

0.008 

0.008 

0.008 

4 

56 

008 

007 

007 

007 

007 

007 

006 

006 

OO6 

006 

006 

3  1 

57 

006 

005 

005 

005 

005 

005 

OO5 

004 

OO4 

OO4 

004 

2  i 

58 

004 

004 

003 

003 

003 

003 

003 

OO2 

002 

002 

OO2 

* 

59 

002 

002 

002 

OOI 

OOI 

OOI 

OOI 

OOI 

OOO 

OOO 

OOO 

O 

m 

lm.O 

Om.9 

Om.8 

Om.7 

Om.6 

Om.5 

Om.4 

Om.3 

Om.2 

Om.l 

Oin.O 

m 

With  Hour-  Angles  greater  than  6h  read  from  bottom,  and  subtract  tabular   1  " 

Log  from  10.000. 

130 


TABLE  XXX.  B. 


- 

Time-Azimutlts  :    I.ou  Tangents  X  and  Y. 

*1 

00 

O'l 

0°.2 

O°.3 

O°.4 

O°.5 

0°.6 

03.7 

0°.8 

O  .9 

-'S 

'Sb'S 

$" 

|a| 

o 

0 

00 

7.242 

7-543 

7-719 

7-844 

7.941 

8.020 

8.087 

8.145 

8.196 

o     ' 

o 

1 

8.242 

8.283 

8.321 

8.356 

8.388 

8.418 

446 

472 

497 

521 

i 

2 

543 

564 

585 

604 

622 

640 

657 

674 

689 

705 

2 

3 
4 

719 
845 

734 
855 

Itl 

761 
876 

774 

886 

786 
896 

799 
906 

811 

822 
924 

834 
933 

3 
1 

5 

8.942 

8.951 

8-959 

8.967 

8.976 

8.984 

8.991 

8.999 

9.007 

9.014 

5 

6 

9.022 

9.029 

9.036 

9-043 

9.050 

9-057 

9.063 

9.070 

076 

083 

6 

7 

089 

095 

102 

108 

114 

119 

I25 

137 

142 

7 

8 

148 

153 

159 

164 

169 

174 

180 

185 

190 

195 

8 

9 

200   i      205 

209 

214 

219 

224 

228 

233 

237 

242 

9 

1O 

9.246   i   9.251 

9-255 

9-259 

9.264 

9.268 

9.272 

9.276 

9.280 

9.285 

10 

11 

289         293 

297 

301 

305 

308 

3^2 

316 

320 

324 

11 

12 

327         33i 

335 

339 

342 

349 

353 

356 

360 

12 

13 

363         367 

370 

374 

377 

380 

384 

387 

390 

394 

13 

14 

397 

400 

403 

406 

410 

4i3 

416 

419 

422 

425 

14 

16 

9.428 
457 

9-431 
460 

9-434 
463 

9-437 
466 

9.440 
469 

9-443 
472 

9.446 
474 

9.449 
477 

9-452 
480 

9-455 
483 

15 
16 

17 

485 

488 

491 

493 

496 

499 

5°4 

507 

5°9 

17 

18 

512 

5H 

517 

519 

522 

525 

527 

530 

S32 

535 

18 

19 

537 

539 

542 

544 

547 

549 

552 

554 

556 

559 

19 

20 

9.561 

9-563 

9.566 

9.568 

9-570 

9-573 

9-575 

9-577 

9.580 

9-582 

2O 

22 

609 

589 
611 

6?] 

593 
615 

595 
617 

598 
619 

600 
621 

602 
624 

604 
626 

21 
22 

23 

628 

630 

632 

634 

636 

638 

640 

642 

644 

647 

23 

24 

649 

651 

653 

655 

657 

659 

661 

663 

665 

667 

24 

25 

9.669 

9.671 

9-673 

9-675 

9-677 

9.678 

9.680 

9.682 

9.684 

9.686 

25 

26 

688 

690 

692 

694 

696 

698 

700 

702 

703 

705 

26 

27 

28 

707 
726 

3 

711 

729 

713 
731 

733 

716 

735 

718 
737 

720 

738 

722 
740 

724 

742 

27 

28 

29 

744 

746 

747 

749 

753 

754 

756 

758 

760 

29 

30 

9.761 

9-763 

9-765 

9-767 

9.768 

9.770 

9-772 

9-774 

9-775 

9-777 

30 

31 

779 

780 

782 

784 

786 

787 

789 

791 

794 

31 

32 

797 

799 

80  1 

803 

804 

806 

808 

809 

811 

32 

33 

gj-2 

814 

816 

817 

819 

821 

822 

824 

826 

827 

33 

34 

829 

831 

832 

834 

836 

837 

839 

840 

842 

844 

34 

35 

9-845 

9.847 

9.848 

9.850 

9-852 

9-853 

9-855 

9.856 

9.858 

9.860 

35 

36 

861 

863 

864 

866 

868 

869 

871 

872 

874 

876 

36 

37 

877 

879 

880 

882 

883 

885 

887 

888 

890 

891 

37 

38 

893 

894 

896 

897 

899 

901 

902 

9°4 

905 

907 

38 

39 

908 

910 

911 

913 

9*5 

916 

918 

919 

921 

922 

39 

40 

9.924 

9-925 

9-927 

9.928 

9-93° 

9-931 

9-933 

9-935 

9-936 

9-938 

40 

41 

939 

941 

942 

944 

945 

947 

948 

95° 

951 

953 

41 

42 

954 

956 

957 

959 

961 

962 

964 

965 

967 

968 

42 

43 

970 

971 

973 

974 

976 

977 

979 

980 

982 

983 

43 

44 

985 

986 

988 

989 

991 

992 

994 

995 

997 

998 

44 

45 

0.000 

O.OO2 

0.003 

0.005 

0.006 

0.008 

0.009 

o.on 

O.OI2 

0.014 

45 

FIRST  CASE  :    Half  sum  of  Polar  Distance  and  Co-latitude  less  than  90°.—  The  j 

sum  or  difference  of  the  Angles  X  and  Y,  according  as  the  Polar  Distance  is  greater 

or  less  than  the  Co-latitude,  is  the  Azimuth. 

SECOND  CASE:   Half  sum  of  Polar  Distance  and  Co-latitude  greater  than  90°.  — 

The  difference  of  the  Angles  X  and  Y  subtracted  from  180°  is  always  the  Azimuth. 

TABLE  XXX.  B. 


131 


Time-Azimuths  :    Log  Tangents  X  and  Y. 

M  . 

N  . 

f! 

OC.O 

0°1 

0°.2 

O°.3 

0°.4 

0°.5 

O°.6 

0°.7 

0°.8 

0°.9 

}l 

o 

45 

o.ooo 

0.002 

0.003 

0.005 

0.006 

0.008 

0.009 

o.on 

0.012 

0.014 

45 

OI5 

017 

018 

020 

02  1 

023 

024 

026 

027 

029 

46 

47 

030 

032 

033 

035 

036 

038 

039 

041 

043 

044 

47 

48 

046 

047 

049 

050 

052 

053 

055 

056 

058 

059 

48 

49 

06  1 

062 

064 

065 

067 

069 

070 

072 

073 

075 

49 

50 

0.076 

0.078 

0.079 

0.081 

0.082 

0.084 

0.085 

0.087 

0.089 

0.090 

50 

51 

092 

093 

095 

096 

098 

099 

IOI 

103 

104 

106 

51 

52 

107 

IO9 

no 

112 

"3 

U5 

117 

118 

120 

121 

52 

53 

123 

124 

126 

128 

129 

132 

134 

136 

137 

53 

54 

139 

140 

142 

144 

147 

148 

150 

152 

153 

54 

55 

0.155 

0.156 

0.158 

O.I  6O 

0.161 

0.163 

0.164 

0.166 

o.i  68 

0.169 

55 

56 

171 

173 

174 

I76 

178 

179 

181 

183 

184 

1  86 

56 

57 

I87 

I89 

191 

192 

194 

I96 

197 

199 

201 

203 

57 

58 

204 

2O6 

208 

209 

211 

213 

214 

216 

218 

220 

58 

59 

221 

223 

225 

226 

228 

230 

232 

233 

235 

237 

59 

60 

0.239 

0.240 

0.242 

0.244 

0.246 

0.247 

0.249 

0.251 

0-253 

0.254 

60 

61 

256 

258 

260 

262 

263 

265 

267 

269 

271 

272 

61 

62 

274 

276 

278 

280 

282 

284 

285 

287 

289 

291 

62 

63 

293 

295 

297 

298 

300 

302 

3°4 

306 

308 

310 

63 

|  64 

312 

3*4 

318 

320 

322 

323 

325 

327 

329 

64 

65 

0.331 

0-333 

0.335 

0-337 

0-339 

0.341 

0-343 

0.345 

0-347 

0-349 

65 

66 

351 

353 

356 

358 

360 

362 

364 

366 

368 

370 

66 

67 

372 

376 

379 

38l 

383 

385 

387 

389 

391 

67 

68 

394 

396 

398 

400 

402 

405 

407 

409 

411 

414 

68 

69 

416 

418 

420 

423 

425 

427 

43° 

432 

434 

437 

69 

70 

0-439 

0.441 

0.444 

0.446 

0.448 

0.451 

0-453 

0.456 

0.458 

0.461 

7O 

71 

463 

465 

468 

470 

473 

475 

478 

481 

483 

486 

71 

72 

488 

491 

493 

496 

499 

504 

507 

509 

512 

72 

73 
74 

515 
543 

545 

520 
548 

523 

526 
554 

5°8 

557 

531 
560 

534 
503 

III 

540 
569 

73 
74 

75 

0.572 

o-575 

0.578 

0.581 

0.584 

0.587 

0.590 

0-594 

0-597 

0.600 

75 

76 

603 

606 

610 

613 

616 

620 

623 

626 

630 

633 

76 

77 

637 

640 

644 

647 

651 

654 

658 

66  1 

665 

669 

77 

78 

673 

676 

680 

684 

688 

692 

695 

699 

703 

707 

78 

79 

711 

7i5 

720 

724 

728 

732 

736 

745 

749 

79 

80 

0-754 

0.758 

0.763 

0.767 

0.772 

0.776 

0.781 

0.786 

0.791 

0.795 

8O 

81 

800 

805 

810 

815 

820 

826 

831 

836 

841 

847 

81 

82 

852 

858 

863 

869 

875 

881 

886 

892 

898 

905 

82 

83 

911 

917 

924 

930 

937 

943 

95° 

957 

964 

971 

83 

84 

978 

986 

993 

1.  00  1 

1.009 

1.016 

1.024 

1.041 

1.049 

84 

85 

1.058 

1.067 

1.076 

1.085 

1.094 

1.104 

1.114 

1.124 

I-I34 

I-I45 

85 

86 

87 

III 

166 
295 

178 

326 

201 

343 

214 

360 

226 
378 

239 
396 

253 

266 
436 

86 

87 

88 

457 

479 

5°3 

528 

554 

582 

612 

644 

679 

717 

88 

89 

758 

804 

855 

980 

2.059 

2.156 

2.281 

2-457 

2.758 

89  j 

90 

-f   00 

9O 

The  Azimuth  is  marked  N  or  S  according  to  the  Latitude,  and  E  or  W  according 

to  the  Hour-  Angle. 

The  Position-  Angle  is  found  by  reversing  the  process  of  the  preceding  Rule  ;  that 
is  to  say,  by  operating  with  the  difference  instead  of  the  sum,  and  the  sum  instead 

of  the  difference,  in  the  two  Cases. 

132 


TABLE  XXXI. 


Time-Azimuths:   Direct  and  Limiting  Values. 

Lat. 

Hour-Angle  West  or  East  of  the  Meridian. 

11 

12 

On  Hori- 
zon. 

Oh 

lh 

2" 

3h 

4h 

5h 

6h 

7" 

8h 

9h 

10 

Az. 

H.A. 

o 

0 
10 
20 
30 
40 
!  5O 
60 
70 
§0 

o 

0 
10 
!  20 
30 
40 
50 
60 
70 
80 

0 

O 

10 

20 
30 
40 
50 
60 
!  70 
80 

Polar  Distance  30°,  or  Declination  60°  of  same  name  as  the  Latitude. 

0 

0.0 

o.o 
o.o 

0.0 
0.0 
0.0 
Indet. 

180.0 
180.0 

8i 

9.6 
H-3 

14-3 

20.1 

34-7 

140.6 

158.3 

o 

16.1 
17.8 

20.6 

25-1 

33-° 

76.9 
H3-9 
137-8 

o 
22.2 
24.1 
27.0 
3L7 

39-0 
51.0 

7°-3 
95-8 
119.2 

o 

26.6 
28.2 
30.7 
34.7 
40.7 
49.9 

63.4 
82.0 
102.5 

o 

29.1 

30.2 
32.1 

35-2 

32i 

46.6 

56-4 
70.1 

87-3 

0 

30.0 

30.4 
31.6 

33-7 
37-o 
41.9 
49.1 
59-4 
73-3 

0 

29.1 
28.9 
29.4 
30.7 
32.9 

364 
41.6 
49.2 
60.  i 

o 

25.8 

25.7 
26.3 
27.7 
30.1 
33-7 
39-2 
47-5 

o 

20.7 
20.9 

21.7 
23.1 
25.6 

29.4 
35-3 

o 

14.6 

14.5 
14.9 

15.7 
17.2 

19.6 

23.4 

7-5 

74 
7.6 

7-9 
8.6 

9-8 
11.7 

0 

o.o 
0.0 
O.O 
O.O 
0.0 

o.o 
o.o 

o 

30.0 

28.4 

22.8 
O.O 

h  in 
6    o 

7" 

836  ] 

12     O 

Polar  Distance  40D,  or  Declination  50°  of  same  name. 

o.o 
o.o 
o.o 
o.o 
o.o 
Iiidet. 
180.0 
180.0 
1  80.0 

o 
12.2 
14.4 

18.1 

25.2 

41.6 

84.2 
132.9 

152.6 
160.8 

22.8 
26.0 

Si 

1% 

107.1 
129.1 
142.3 

0 

30-7 

46.2 

57-i 
72.4 

9L3 

1  1  O.O 

124.7 

36.0 

38.5 

42.4 

47-9 

79-4 
94.1 
108.3 

o 

39-0 
40.6 

43-i 
47-o 

52-3 
59.6 
69.0 
80.4 
92.8 

o 
4O.O 
40.4 
41,7 
44.1 
47.6 
52.6 

59-2 
67.8 
78.3 

0 

39-o 
38.4 
38.6 

39-7 
41.8 

45-o 
49-7 
56.0 
64.4 

33-8 
34-o 
35-i 
37-o 
40.1 
44.6 
Si-i 

0 

270 

27-3 
28.4 

30-3 

33-4 
38.0 

o 
18.8 

1  8.8 

19-3 
20.4 
22.3 
25-3 

o 
9-7 

9.6 
9.8 

IO.2 
II.  I 
12.6 

0 

o.o 
0.0 
O.O 
O.O 

o.o 
o.o 

o 
40.0 

38.9 

354 

27.8 
o.o 

h  m 
6    o 

649  i 

743 
850 

I2~0 

Polar  Distance  50°,  or  Declination  4O°  of  same  name. 

0 

0.0 
0.0 

o.o 

0.0 

Indet. 
1  80.0 
1  80.0 
180.0 
180.0 

0 

I7.I 
21.4 

29-5 
46.7 
85.2 
127.8 
148.2 

157-4 
162.2 

30.8 
36.5 
45-4 

£ 

103.9 
123.5 

136.5 
144.8 

0 

40.1 
45-2 

52-3 
62.2 

75-1 

90.2 

105-3 
II8.I 
127.9 

0 

45-9 
49-5 
54-5 
61.2 
69.6 
79-8 
90.9 
101.9 
1  1  1.8 

49°o 
5i.o 

K 

|j 
87.4 
96.4 

o 
5O.O 

5°-4 
51-7 

54-o 
57-3 
61.7 
67.2 
74-o 
81.7 

0 

49.0 

48.0 

47-8 
48.4 
50.0 
52.6 

56.3 
61.2 
67.5 

o 

42.1 

41-6 
41.9 

43-2 
454 
48.8 

53-6 

0 

32.8 

33-2 
344 
36.6 
40.0 

22^6 

22.6 
23.2 
24.4 
26.6 

o 
1  1.6 

II.4 

1  1.6 

12.2 
13-3 

o 

O.O 

O.O 

o.o 
o.o 
o.o 

o 
50.0 
49-2 
46.8 
42.1 

32-9 
O.O 

h  m 
6   o 

634 

711 

756 
859 

12    0 

Polar  Distance  60°,  or  Declination  30°  of  same  name. 

0 

0 
10 
20 
30 
40 
5O 
60 
70 
j  80 

0 

0 
10 
20 
30 
40 
50 
60 
70 
80 

o 
0.0 
0.0 
0.0 
Ind't. 
iSo.O 
1  80.0 
ISO.O 
1  80.0 
1  80.0 

0 

24.1 
32.8 
5O.6 

86.2 
124.6 
144.9 

154-7 
1  60.0 
163.1 

40.9 
50.1 
63.8 
82.4 
102.9 
120.3 
132.7 
140.9 
146.4 

0 

50.8 

57.8 
67.0 

78.3 
91.0 
103.6 
114.6 
123.4 
130.1 

o 

60.9 
66.8 

73-9 
82.1 
90.8 
99-5 
107-5 
114.4 

0 

59-i 

64.8 
69.0 
74.1 
79-8 
86.2 
92.7 
99.1 

,  ° 
60.0 

60.4 
61.5 
63-4 
66.1 
69.6 

73-9 
78.8 

84.3 

0 

59-i 
57-6 
56.8 
56.9 

57-8 

59-5 
62.0 
65.5 
69.8 

0 

49.1 
48.6 
49.0 
50.2 
52.4 

55-6 

o 
37-8 

38.1 

394 
41.6 

o 

25.8 

25-7 
26.3 
27-7 

o 

T3-! 

12.9 
13.2 
13-8 

o 
o.o 

O.O 
0.0 

o.o 

„  o 
6O.O 

59-5 
57-9 
54-7 
49-3 
38.9 
o.o 

/'   in 
6   o 
623 
649  i 
7i8 
756 
854 

12     0 

Polar  Distance  70°,  or  Declination  20°  of  same  name. 

o 
0.0 
0.0 
Indet. 
iSo.O 
180.0 
iSo.O 
iSo.O 
iSo.O 
iSo.O 

0 

354 
53-6 
874 
123.0 
142.9 
152.9 
.158.4 
161.7 
163.8 

o 

53-9 

& 

I03-3 
119.1 
130.7 
138.6 
144.0 
147.6 

62.7 
71.6 
81.9 

93-  ' 
104.0 

II3-5 
121.3 
127.4 
131.8 

6h 

72.6 
78.8 

85-7 
92.8 

99-8 
106.2 
111.7 
116.4 

o 

69.4 

72.0 

75-3 
79.1 

834 
87.9 

92-5 
97.0 

IOI.2 

o 

70.0 

70.3 
71.1 

72.5 

744 
76.8 

79-7 
82.9 
86.4 

o 

69.4 
67.3 

66.0 
65.3 
65-3 
65.9 
67.2 
69.2 
71.8 

o 
55-3 
54-5 

54-6 

55-5 
57-3 

o 

41.7 

41.9 

43-o 

o 

28.2 
28.1 
28.6 

o 
J4'3 
I4.I 
14-3 

o 
o.o 

O.O 
O.O 

0 

7O.O 
69.7 

68.6 
66.7 
63-5 
57.9 
46.8 
o.o 

h  in 
6    0 

*ll\ 

6  30  ; 

649 
711  j 

743 

836 

12     O 

The  Azimuths  in  smaller  figures  answer  to  actual  depressions  of  the  object  below 
the  True  Horizon,  and  are  used  for  differences  with  the  adjacent  Azimuths. 

TABLE  XXXI. 


133 


Time-Azimuths:    Direct  and  Limiting  Values. 

Lat. 

Hour-Angle  West  or  East  of  the  Meridian. 

On  Hori- 
zon. 

01 

1" 

2h 

3h 

4h 

5h 

6" 

7h 

8" 

9h 

10h 

llh    12h 

Az. 

H.A. 

Pol.  Dist.  80°,  or  Declination  10°  of  same  name  as  the  Latitude. 

0 

O 
1O 
i  20 
30 
!  4O 
50 
60 
70 
80 

o 

O 
10 
20 
30 
4O 
50 
6O 
70 
8O 

o 

0 
10 

20 
3O 
40 
50 
6O 
1  70 
80 

o 

0 
10 

20 
30 
!  40 
i  50 
60 
70 

o 

0 
10 
20 
30 
40 
50 
60 

0 

O 
10 
20 
30 
40 
j  50 

o 
0.0 
Indet. 
180.0 
l8o.O 
l8o.O 
l8o.O 
l8o.O 
l8o.O 
l8o.O 

°8 

122.5 
141.9 
152.0 
157-6 
160.9 
163.0 
164.3 

o 

70.6 

87.3 

104.6 

"9-3 
130.1 

137-7 
142.9 
146.4 
148.7 

o 

76.0 

85.9 

96.2 
105.9 
114-3 

I2I.2 
126.6 
I30-5 

J33-3 

78°5 
84-3 
90.4 
96.4 

IO2.2 
!07-3 

in.  7 

II5-3 
118.1 

o 

79-7 
82.4 
85.4 
88.6 
91.9 

95-° 
98.0 
100.7 
103.1 

80.0 
80.2 
80.6 
81.3 
82.3 

83-5 
85.0 
86.6 
88.3 

o 
79-7 
77.2 

75-3 
73-7 
72.7 
72.1 
72.1 
72.6 

73-5 

o 

58.9 

58.5 
58.9 

o 
49-3 
44.2 

o 
29.8 

29-5 

o 

15-0 

14.8 

o 
o.o 

0.0 

o         ft   in 

80.0     6   o 
79.8     6    7 

79-3     6  1S 

78.4  i    6  23  i 
76.9     6  34  i 
74.3     649  j 
69.7     711  ! 

59-5     7  56  I 

O.o  ;  12    o 

Pol.  Dist.  90°,  or  Declination  0°. 

Indet. 
180.0 
1  8o.O 
180.0 
180.0 
180.0 
180.0 
180.0 
180.0 

o 
9O.O 
123.0 
I4I.9 
I5I.8 

1574 
160.7 
162.8 
I64.I 
164.8 

o 

90.0 

106.7 

120.6 
130.9 
138.0 
143.0 
146.3 
148.4 
149.6 

o 

90.0 

99-9 
108.9 
116.6 
122.7 

127-5 
130.9 
133.2 
134.6 

0 

90.0 

95-7 

IOI.2 
I06.I 
IIO.4 

"3-9 
116.6 
118.5 
119.6 

0 

90.0 

92.7 

95.2 

97.6 

99-8 
101.6 
103.1 
104.1 
104.8 

0 

90.0 
90.0 
90.0 
90.0 
90.0 
90.0 
90.0 
90.0 
90.0 

o 
90.0 

87.3 
84.8 
82.4 

80.2 

78.4 

76.9 

75-9 
75.2 

0 

9O.O 
90.0 
9O.O 
9O.O 
9O.O 
9O.O 
9O.O 
9O.O 
9O.O 

h  m  \ 
6    0 
6    0 

6   o 
6   o 
6   o 
6   o 
6   o 
6   o 
6   o 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name. 

0 

180.0 
180.0 
180.0 
1  80.0 
180.0 
180.0 
1  80.0 
180.0 
180.0 

0 

124.3 
142.8 

152-5 
157.8 

161.1 
163.1 
164.4 
165.0 

165.2 

0 

109.4 
123.0 
132.7 

139-5 
144.1 
147.2 
149.2 
150.2 
*5°-5 

o 

104.0 

112.7 

I2O.O 
125.6 
129.8 
132.8 
134-7 
135-7 
135-8 

0 

101.5 
106.7 

111.3 

114.9 
117.8 
119.8 

121.  1 
I2I-5 

121.  1 

o 

100.3 

102.8 

104.8 

106.3 
107.3 
107.8 

107.9 
107.5 

0 

IOO.O 

99.8 

99-4 
98.7 

97-7 
96.5 

o 
100.3 

- 

o 
IOO.O 
100.2 
IOO-7 

101.6 

103.1 

105.7 
110.3 

120.5 
180.0 

k  m 
6  o 
553 
545 
537 
526 

5" 

449 
4  4 

0    0 

Pol.  Dist.  110°,  or  Declination  20°  of  contrary  name. 

0 

180.0 
180.0 

180.0 
180.0 
180.0 
180.0 
1  80.0 
180.0 

o 

144.6 

153-8 

159.0 

162.0 
164.0 
165.1 

165.7 

165.9 

o 
I26.O 

135-5 
I4I.9 
146.2 
I49.I 
150.9 
I5I.8 
I5L9 

o 

"7-3 

124.2 
129.5 

133-4 
136.1 
137.6 

138-3 

138.1 

0 

1  12.8 

117.2 

120.6 

123.1 

124.7 
125.5 

125.4 

o 

110.6 
112.7 

114.0 
114.7 
114.7 
114.1 

0 

IIO.O 

109.7 
108.9 
I07-5 

0 

1  10.6 

0 

1  1  0.0 

110.3 
111.4 

II3-3 
116.5 

1  22.  1 

133-2 
I  So.O 

h  m 

6   o 

545 
530 
5" 
449 
417 
324 
o   o  ! 

Pol.  Dist.  120°,  or  Declination  30°  of  contrary  name. 

i8o°o 
1  80.0 
180.0 
180.0 
180.0 
1  80.0 
180.0 

'$ 

160.6 

'63.5 
165.2 
166.3 
166.9 
167.0 

I39-I 
145.2 
149.2 
I5I.8 

1534 
154.2 

154-3 

0 

129.2 

134.4 
138.0 
140.4 
141.7 

142.2 
141.9 

0 

123.7 
I27.I 
129.5 
130.9 

1314 

i3i.i 

0 

120.9 
122.4 

123.2 

123.1 

122.3 

0 

120.0 
119.6 
118.5 

0 

120.9 

0 

120.0 
120.5 
1  22.  1 

125.3 
130.7 
I4I.I 

l8o.O 

h   m 

6   o  1 
537 
5"  i 
442  | 
4   4 
3   6 
o   o 

Pol.  Dist.  130°,  or  Declination  40°  of  contrary  name. 

h     TO~j 

6   o  ! 
526 
449 
4   4 
3    * 
o  o 

180.0 
180.0 
180.0 
180.0 
180.0 
180.0 

162.9 
165.4 
167.0 
167.9 
168.4 

168.6 

149.2 
152.9 
155-3 
156-7 
157-4 
157-4 

139.9 
143-3 
145.6 
146.8 
147.2 

146.8 

134-1 

136-5 
137-9 
138.4 
138.1 

o 

131.0 
132.0 

132.2 

131.6 

0 

130.0 

129.6 

o 
131.0 

o 
I3O.O 
130.8 
133-2 

*37-9 
147.9 
180.0 

134 


TABLE  XXXII. 


Position-Angles  for  Direct  and  Limiting  Time-  Azimuths. 

Lat. 

Hour-  Angle  West  or  East  of  the  Meridian. 

Lat. 

Oh 

lh 

2h 

3h 

4h 

5h 

6h 

7h 

8h 

9h 

BO 

11" 

12h 

o 

0 
10 
20 
30 
40 
50 
60 
•70 
80 

o 

O 
10 
20 
3O 
40 
50 
60 
70 
8O 

0 

0 
10 

20 
3O 
40 
50 
60 
7O 
80 

o 

0 
10 
20 
30 
40 
50 
60 
1  70 
80 

o 

0 
10 
20 
30 
40 
50 
60 
70 
80 

PoL  Dist.  3O°,  or  Declination  60°  of  same  name  as  the  Latitude. 

o 

0 
10 

2O 
30 
40 
50 
GO 
7O  i 
80 

o 

0 
10 
20 
30 
40 
5O 
60 
70 
8O 

i8o!o 
180.0 
180.0 
180.0 
180.0 
180.0 
Indet. 
O.O 
0.0 

162.8 
160.9 
158.4 

1547 
148.2 
132.9 
83-5 
25-7 
7-4 

146.3 
142.9 
138-7 
132-7 
123-5 
107.1 
76.9 
38.7 
13-5 

I32°? 
126.6 

121.  2 
II4.6 
105.3 
91-3 
70-3 
42.9 
17-6 

n6°6 
111.7 
106.2 

99-5 
80.9 

79-4 
63-4 
42.6 
19.8 

0 

103.1 
98.0 
92.5 

86.2 
78.6 
69.0 

56-4 
40.0 
20.3 

0 

90.0 

85.0 

79-7 
73-9 
67.2 

59-2 
49.1 
36.1 
19.4 

0 

76.9 

71.1 
67.2 

62.0 

56.3 

49-7 
41.6 
31.2 
17-5 

o 
59-o 

54-6 
50.2 

45-4 
40.1 

33-7 
25-6 
14-8 

0 

41.7 

38.1 

34-4 
30-3 
25.6 
19.6 
n.6 

0 

28.2 

25-7 
23.2 

20.4 

17.2 

*3-3 
7-9 

0 

14.3 

12.9 

11.6 

IO.2 

8.6 

6.7 
4.0 

o 

O.O 

0.0 
O.O 
O.O 
0.0 
O.O 
O.O 

Pol.  Dist.  40°,  or  Declination  50°  of  same  name. 

i8o°o 
180.0 
180.0 
180.0 
180.0 
Indet. 
O.O 
O.O 
0.0 

o 
160.7 

157-6 
152.9 
144.9 
127.8 
84.2 

34-7 
14.2 

5-i 

143.0 

137-7 
130.7 
120.3 
103.9 
78.4 
48.0 
24.4 
9-5 

o 

127.4 

I2I.2 

"3-5 
103.6 
90.2 
72.4 
51.0 
30.0 

12.8 

o 

"3-9 
107.3 
99.8 
90.8 
79-8 
66.1 

49-9 
32.1 
14.9 

o 

101.6 

95-o 
87.9 

79-9 
70-5 
59-6 
46.6 

31-7 
15.6 

o 
90.0 

83.5 

76.9 
69.6 

61.7 
52.6 
41.9 
29-5 
15-3 

o 

78.4 

72.1 

65-9 

59-5 
52-6 

45-° 
36-4 
26.2 
14.1 

o 
54-5 
48.9 
43-2 
37-o 
30.1 

22.0 
12.  1 

0 

378 

2£ 

22.1 

17.0 
9.6 

0 

25.8 

22.6 
19-3 

I5i 

u.6 
6.6 

0 

*3-i 
II.4 
9.8 

8.0 
5-9 
3-4 

o 

0.0 

0.0 
O.O 
0.0 
0.0 
O.O 

Pol.  Dist.  50°,  or  Declination  40°  of  same  name. 

0 

0  I 
10 
20 
3O 
40  ! 
50 
60 
7O 
80 

i8o°o 
180.0 
180.0 
180.0 
Indet 
0.0 
O.O 
0.0 
O.O 

o 

157-4 

152.0 

142.9 

124.6 

85.2 

41.6 

20.1 

9-9 

4.0 

138°© 
130.1 
119.1 
102.9 
80.2 
54-6 
33-o 
17.9 

7-5 

o 
122.8 

II4-3 
104.0 
9I.O 

75-i 
57-i 
39-o 
23-2 
10.3 

o 

110.4 

IO2.2 
92.8 
82.1 
69.6 

55-7 
40.7 

25.9 

I2.I 

99.8 
91.9 
834 

$ 
§1 

26.5 
13.0 

o 
9O.O 
82.3 

74-4 
66.1 

57-3 
47.6 

37-o 
25-4 
13.0 

o 

80.2 
72.7 

65.3 

57.8 

50.0 

41.8 

32-9 
23.0 

12.1 

o 
55-3 
48.6 
41.9 

35-o 
27.7 
19.6 
10.5 

0 

32-8 

27-3 
21.6 

8.4 

o 

22.6 

18.8 
14.9 
10.6 
5-8 

o 
it  6 

9-6 

7.6 

5-4 
3-o 

O.O 

O.O 
O.O 
O.O 
O.O 

Pol.  Dist.  60°,  or  Declination  30°  of  same  name. 

0 

0 
10 
20 
3O 
40 
50 
OO 
7O 
80  \ 

o 

O 

10 

20 
30 
40 
50 
6O 
70 
80 

rt  o 

1  80.0 
180.0 
1  80.0 

Indet. 
0.0 
O.O 
O.O 
0.0 
0.0 

151.8 
141.9 
123.0 
86.2 
46.7 
25.2 

14-3 
7-8 
3-3 

130.9 
"9-3 
103-3 
82.4 
59-6 
39-8 
25.1 
14.4 
6.4 

n6°6 
105.9 
93-i 
78.3 
62.2 
46.2 

3i-7 
19.2 
8.8 

o 

106.1 
96.4 
85.7 

73-9 
61.2 
48.0 
34-7 

22.1 
10.5 

o 

7*1 

69.0 

58.3 
46.9 

35-2 
23-2 
11.4 

o 
9O.O 
8l.3 
72.5 

634 
54-0 
44.1 

33-7 
22.8 

"•5 

8,!4 
73-7 
65.3 
56.9 
48.4 

39-7 
30.7 

21.  1 

10.8 

o 

49.1 
41-5 
34-o 
26.3 
18.2 
9-5 

o 

27.0 

20.7 

14-5 
7.6 

o 
18.8 

14-5 
10.  1 

5-4 

o 
9-7 

7-4 
5-2 
2.7 

o 

O.O 

O.O 
0.0 
0.0 

Pol.  Dist.  70°,  or  Declination  20°  of  same  name. 

i8o°o 
1  80.0 
Indet. 

0.0 
O.O 
O.O 
0.0 
0.0 
0.0 

141.9 
122.5 
87.4 
50.6 

33 
tj 

3-o 

0 

120.7 
104.6 

84.8 
63.8 

45-4 
31-3 

20.6 

12.3 

5.7 

io8°9 
96.1 
81.9 
67.0 

III 

27.0 
1  6.8 
7-9 

o 
IOI.2 
90.4 
78.8 

66.8 
54-5 
42.4 
30-7 
19.8 

9-5 

o 

95-2 
85-4 

S3 

54-i 
43-i 
32.1 

21.2 
10.5 

o 
90.0 

80.6 

67;.-5 

Si 

31.6 

21.2 
10.6 

84°8 
75«3 
66.0 
56.8 

47-7 
38.6 
29.4 
19.9 

10.  1 

0 

42.1 

33-8 
25-7 
17-5 
8.9 

0 

20  7 

14.1 

7-2 

I4°6 

9-8 
5-i 

o 
7-5 

8 

O.O 

O.O 
O.O 

Position-  Angles  in  smaller  figures  correspond  to  depressions  of  the  object  below 
the  True  Horizon. 

I 

TABLE  XXXII. 


135 


Position-Angles  for  Direct  and  Limiting  Time-Azimuths. 

Lat. 

Hour-  Angle  West  o 

r  Eas 
6h 

b  of  the  Meridian. 

T  •>*•   ' 

Oh        lh 

2h 

3h 

4h 

-1 

7h 

8h 

9h 

10'' 

llb     12h 

j-iat. 

o 

O 
10 
20 
30 

4O 
50 
60 

70 

80 

o 

0 
10 

20 
30 
40 
50 
60 
70 
80 

0 

0 
10 

20 

30 
40 

50 
60 
7O 

80 

o 

0 
10 

20 

30 
40 
50 
60 
70 

0 

0 
10 
20 
30 
40 
50 
60 

0 

0 
10 
20 
30 
40 
5O 

Pol.  Dist.  80°,  or  Declination  10°  of  same  name  as  the  Latitude. 

o    ; 

0 
1O 

20  ! 
3O 
40  i 
50 
60 
70  ! 
80  ! 

i8o?o 
Indet. 
O.O 
O.O 
O.O 
O.O 
O.O 
O.O 
O.O 

0 

123.0 

88.7 

53-6 

32-9 
21.5 
14.4 
9-5 
5-8 
2.7 

o 

106.7 

87.3 

67.4 

50.1 
36.5 

26.0 

17.8 

II.  I 

5-3 

99-8 
85-9 
71.6 
57.8 
45-2 
33-9 
24.1 

'5-3 

7-4 

o 

95-7 
84-3 
72.6 
60.9 

49-5 
38.5 
28.2 

18.3 
8.9 

o 

92.7 

82.4 

72.0 

61.5 
51.0 

40.6 

30.2 
19.9 

9-9 

o 

90.0 

80.2 

70.3 

60.4 

50-4 
40.4 

3°-4 
20.3 

IO.2 

o 

87.3 

77-2 
673 
57-6 
48.0 
38-4 
28.9 
194 

9-7 

0 

25.8 

17.1 

8.7 

0 

140 
7-i 

o 
9-9 
5.0 

2.6 

0 
0.0 

O.O 

Pol.  Dist.  90°,  or  Declination  0°. 

o 

O 

10  | 

20 
30  i 
40 
50 

60  ,' 
70 

8O 

o 

0 
10 
20 
3O 
40 
50 
60 
70 
80  i 

Indet. 
0.0 
O.O 
0.0 
O.O 
0.0 
O.O 
O.O 
0.0 

0 

90.0 
55-7 
35-4 
24.1 
17.1 

12.2 

8.5 

5-4 

2.6 

o 
90.0 
7O.6 

54-0 
40.9 
30.8 
22.7 
16.1 
10.3 
5-o 

o 

90.0 
76.0 

62.8 
50.8 
40.1 

30.6 

22.2 
14.4 

7-i 

o 

90.0 

78.5 
67.2 
56.3 

45-9 
36.0 
26.5 

17-5 
8.7 

o 

90.0 

79-7 
69.4 

59-1 
49.0 
39-o 
29.1 
19.4 
9-7 

o 
9O.O 
80.0 
70.0 
6o.O 
5O.O 
4O.O 
30.0 
2O.O 
1  0.0 

0 

90  o 

79-7 
69.4 

59-1 
49.0 
39-o 
29.1 
19.4 
9-7 

i 
| 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name. 

o 
O.O 
O.O 
0.0 
O.O 
0.0 
O.O 
O.O 
O.O 
O.O 

o 

57-o 
37-i 
26.1 

19-3 
14.6 
10.9 
7.8 

5J 

o 

73-3 
57-o 
44-5 
34-8 
27.1 
20.7 

I5-* 

9-9 

5-o 

80?! 

67.2 

55-7 
45-6 
36.7 
28.6 

21.  1 

14.0 
7-i 

8*3 

m 

52.8 

43-4 
34-5 
25.8 
17.2 

8.7 

87-3 
77-2 
67-3 

57-6 
47-9 
38.5 

28  9 
19-3 

0 

90.0 

80.  i 
70-3 
60.4 
50-4 
40.4 

o_ 
92.7 

Pol.  Dist.  110°,  or  Declination  20°  of  contrary  name. 

o 
0.0 
O.O 
O.O 
O.O 
0.0 
O.O 
O.O 
0.0 

o 
38.1 

27-5 
21.0 
I6.5 
I3.0 
10.  1 

7-5 

5-1 

0 

59-4 
47.2 

38-1 
30.8 
24.7 
19.4 
14.6 

9-9 

0 

71.1 

60.0 

5°-4 
42.0 

34-4 
27.4 
20.7 
14  i 

78.8 
68.8 
59-4 
50-5 
42.0 

33-8 
25-7 

84.8 

75-3 
66.0 
56.8 

47-8 
38-6 

0 

90.0 

80.6 
71.1 

61.5 

0 

95-2 

o   ! 
0 

10  ! 
20  i 

30  i 
40  ! 
50 
6O 

70  ! 

o   i 

o 

10  ! 
20  ! 

30  I 
40  1 
50 
6O 

Pol.  Dist.  120°,  or  Declination  30°  of  contrary  name. 

o 
O.O 
0.0 
0.0 
O.O 
O.O 
O.O 
0.0 

2& 
22.1 
I7.9 
14.7 

i2.i 

9-7 

7-4 

o 
49.1 

40-5 

%.l 

S3 

14-5 

o 

63.4 

54-4 
46.6 

39-6 
33-2 
27.0 
20.9 

o 

73.9 
65.1 
56.9 
49.1 

41.6 

34-o 

82.°4 
73-5 
65-3 
56.9 
48.4 

o 

90.0 

81.3 

72.5 

o 

97.6 

Pol.  Dist.  130°,  or  Declination  40°  of  contrary  name. 

o 
0.0 
O.O 
0.0 
0.0 
0.0 
0.0 

o 
22.6 
18.9 

1  6.0 

13-7 

n.6 

9.6 

o 

41.9 

35-9 
3°;9 
26.6 

22.6 

18.8 

0 

57-2 
50.2 

43-9 
38.2 
32-8 
27.4 

o 
69.6 
62.2 

48.6 
41.9 

80.2 
72.7 
65-3 
578 

o 

90.0 

82.3 

o 
99.8 

o 

0 
10 
20 
30 
40 
5O 

136 


TABLE  XXXIII 


Altitudes  for  Direct  and  Limiting  Time-Azimuths. 

Lat. 

! 

0 

0 
10 
20 
30 
40 
5O 
60 
70 
8O 

0 

0 

10 

20 
30 
40 

'  50 
60 
i  70 
80 

0 

0 
10 

20 

!  3O 
40 
50 
6O 
70 
80 

o 

0 

10 

20 

I  30 
140 
50 
60 
i  70 
80 

o 

0 
10 
20 
30 
40 
50 
6O 
70 
80 

Hour-Angle  West  or  East  of  the  Meridian. 

Lat. 

6b 

7h 

8h 

9h 

10h 

llh 

8-2 

PoL  Dist.  30°,  or  Declination  60°  of  same  name  as  the  Latitude. 

o 

O 

10 

20 
30 
40  ! 
50 
60 
70 
80 

0 

O 

10 

20 
30 
40 
50  i 
6O 
70  ! 
8O 

0 

30.0 
40.0 
50.0 
60.0 
70.0 
80.0 

90.0 

80.0 

70.0 

0 

28.9 

38.7 

48.6 

58.4 

67.9 
76.9 
82.3 
78.2 

69-5 

0 

25-5 

35-0 
44-6 
53-9 
62.7 
70.4 
75-i 

m 

0 

20.7 
30.0 

38.9 
47.5 

55.8 
62.9 
68.0 
69.1 
66.0 

0 

14.4 

23.4 
32.1 

40.5 

48.4 

55-5 

64!i 
637 

0 

,£? 

24.7 

33-o 
40.9 

48.3 
54-6 
59-1 
61.1 

0 

0.0 
8-7 
17-3 

25-6 

33-8 
41.6 

48.6 

54-5 
58.5 

0 

7-3 
1.2 
1  0.0 
I8.7 
27.2 

35-5 
43-3 
50.3 

56.1 

o 
5-5 

3-3 

12.4 
21.4 
30.2 

38.7 
46.7 

54-0 

o 
i-7 

16^6 
25.8 

34-9 
43-8 
52.3 

o 
6.6 

3-2 

I3-I 
22.7 

33 

5i-i 

o 

8.9 

0.9 

10.8 
20.7 
30.6 
40.4 
50.2 

o 

IO.O 

O.O 
IO.O 
2O.O 
30.0 
4O.O 
5O.O 

Pol.  Dist.  403,  or  Declination  50°  of  same  name. 

o 

40.0 

50.0 
60.0 
70.0 

80.0 

90.0 
80.0 
70.0 
60.0 

o 
38.2 
48.1 

57-7 
67.0 

SI 

76.8 
68.7 

59-6 

33-8 
42.9 
51.8 

59-9 
66.8 
70.8 
70.4 
65.5 
58.3 

o 

27.1 

35-5 
43-6 
5°-9 
57-2 
61.5 
62.9 
61.1 
56.4 

i8.°8 
26.7 
34-3 
414 
47-6 
52.5 
55-5 
56.1 
54-1 

o 

9-7 
17.2 
24.6 
31.8 
38.3 
43-9 
48-3 
51.0 
51.6 

o 
O.O 

7.7 
15.2 
22.5 
29-5 

35-9 
41-5 
46.1 
49.0 

o 
9-7 
1.8 

6.0 
13.8 
21.4 
28.7 
35-5 
41-5 
46.5 

0 

5-9 
14-3 

22.2 
30.2 

37-6 
44-3 

o.7 
8.1 
17.1 

25.8 

34-3 
42.6 

5-8 
3-9 
13-4 
22.7 
32.0 
41.2 

8.6 
0.9 

10.8 
20.7 
30.5 
40.3 

IO.O 

0.0 
IO.O 
2O.O 
30.0 
4O.O 

Pol.  Dist.  50°,  or  Declination  40°  of  same  name. 

0 

0  I 

10 

20 
30 
40 
50! 
60 
70 
8O 

0 

50.0 
60.0 
70.0 

80.0 

90.0 

80.0 
70.0 
60.0 
50.0 

0 

47-7 
57-2 
66.2 
74-2 
78.5 
75.6 
67-9 
59-o 
49-6 

o 

41.5 
49.9 

m 

67.1 
66.7 
62.7 

56.2 
48.4 

32.8 
40.2 
46.8 
52.2 
55-9 
57-2 

52l 
46.6 

o 

22.5 
29-3 

35-4 
40.8 

45  -° 
47-6 
48.4 

47.3 
44-4 

o 

11.4 
17.9 

24.0 

29-5 

34-4 
38.3 
41.0 
42.2 
41.9 

o 
0.0 

6-4 
12.7 
I8.7 
24-4 
29-5 

33-8 
3M 
39-3 

0 

11.4 

4.8 

2.1 

8.8 
15.2 
21.4 
27-3 
32-4 
36.8 

o 
8.0 

o.7 
6.9 

14-3 
21.4 

28.2 
34-5 

o 

O.I 

8.2 

16.6 
24.8 
32.6 

o 
5-8 

3-7 
'3-i 

22.2 
31.2 

o 
8.8 

0.6 
10.6 
20.5 
3°-4 

o 

IO.O 

O.O 
IO.O 
2O.O 
30.0 

Pol.  Dist.  60°,  or  Declination  30°  of  same  name. 

0 

o; 

10 

20 

30 
40 
50 
6O 
70 
80 

o 

0 
10 

20 
30 
40 
50 
6O 
70 
80 

o 

60.0 
70.0 
80.0 

70.0 
60.0 
50.0 
40.0 

56.8 
65.6 
73-2 
77.0 
74.2 
67.0 

58.3 
49.1 
39-6 

48^6 
55.6 
61.1 
64.1 
63.6 
59-9 
53-9 
46.6 

38.5 

37^3 
43-6 
48.3 
5i-3 
52-2 

50-9 
47-7 
42.8 
36.8 

o 

25.6 

3°-9 

$ 

40.8 
41.4 

40-5 
38.2 

34-6 

o 

12.9 
17.9 

22.4 
26.3 
29.5 
31.8 

33-o 
33-i 
32-1 

o 
O.O 

5-o 
9-9 

14-5 
1  8.8 
22.5 

25-7 
28.0 

29-5 

o 
12.9 

7.8 

2-5 

3-° 

8.6 
13-8 
18.7 
23-2 
20.9 

o 
7.0 
0.6 

5-9 
12.4 
18.8 
24.6 

o 
0.8 

7-2 

I5-I 

22.5 

0 

6.0 

3-o 

12.4 
21.4 

0 

9.0 

0.5 
10.3 

20.  2 

o 

IO.O 

O.O 
IO.O 
20.0 

Pol.  Dist.  70°,  or  Declination  20°  of  same  name. 

0 

70.0 
80.0 

90.0 

80.0 
70.0 
60.0 
50.0 
40.0 
30.0 

6s.-, 

72.4 
75-9 
73-2 
66.2 

57-7 
48.6 
39-2 
29.6 

0 

54-5 

l?i 

61.8 
61.1 

57-5 
5i-7 

$1 

28.6 

41^6 
45.6 
47-8 
48-3 
46.8 
43-6 
3*9 
33-3 
26.9 

0 

28.0 
31-5 

33-9 
35-3 
35-4 
34-3 

1:1 

24-7 

o 

14.1 
17.4 

20.2 
22.4 
24.0 

24-7 
24.7 

23-9 
22.3 

o 
0.0 

3-3 
6.7 
9.9 
12.7 
15-2 
17.2 
18.7 
19.7 

0 

14.1 
10.4 

6.4 

2.2 

2.0 

6.1 

10.  1 

13.8 
17.1 

o 
8.1 

2.4 

3-5 
9.2 
14.8 

o 
1.8 

,ij 

o 

6.2 

2.5 

"•3 

o 
9.2 

0.4 
10.3 

o 

IO.O 

0.0 
IO.O 

The  Altitudes  in  smaller  figures  are  depressions  below  the  True  Horizon  ;  that  is, 
negative  Altitudes. 

TABLE  XXXIII. 


137 


Altitudes  for  Direct  and  Limiting  Time-Azimuths. 

^ 
Lat. 

Hour-  Angle  West  or  East  of  the  Meridian. 

Lat. 

Oh 

lh 

2h 

3h 

4h 

5h 

6h 

7h 

8* 

9h 

lOh 

llh 

12h 

Pol.  Dist.  80°,  or  Declination  10°  of  same  name  as  the  Latitude. 

o 

O 
10 
20 
30 
40 
50 
60 
70 
80 

c 

0 
10 
2O 
30 
40 
50 
60 
70 
80 

o 

0 

10 

20 
30 
£0 
50 
6O 
70 
§0 

o 

80.0 

90.0 

80.0 

70.0 
60.0 
50.0 
40.0 

30.0 

20.0 

0 

72.1 

75-2 

Hi 

57-2 
48.1 
38.8 
29.2 
19.6 

58-°5 
60.5 

594 
55-6 
49.9 
42.9 
35-2 
27.1 
18.6 

o 

44.1 

45.7 

45  -6 
43-6 
40.2 

35-5 
29.9 

23-7 
17.0 

o 

29-5 
31.0 

31.5 

3°-9 
29-3 
26.7 

234 
19.4 
14.8 

0 

14.8 

16.3 

17.4 
17.9 
17.9 

'7-3 
16.1 

H-5 
12.4 

0 

O.O 

1.8 

3-3 
4.9 

6.4 
7.6 
8.6 

9-9 

9.8 

0 

14.8 

12.8 

10.4 

7-7 
4-9 
1.8 

°-3 

4.2 

7-3 

o 
5-5 
O.O 
4.9 

0 

5-8 

3-o 

0 

7-1 
2.0 

o 
9-4 
0.2 

0 
IO.O 

0.0 

Pol.  Dist.  90°,  or  Declination  0°. 

o 

0 
10 
20 
30 
40 
50 
60 
7O 
80 

o 

90.0 

80.0 
70.0 
60.0 

5°»° 
40.0 
30.0 

20.0 
10.0 

0 

75-o 
72.0 
65.2 
56.8 
47-7 
384 
28.9 

19-3 
9-7 

,  ° 
60.0 

58.5 

54-5 
48.6 
41.6 
33-8 
25-7 
17.2 
8.7 

0 

45-° 
44.1 
41.6 

37-7 
32.8 
27.0 
20.7 
14.0 
7.0 

o 
30.0 

29-5 
28.0 

25.7 
22.5 
18.7 

14-5 
9-9 
5-i 

o 

15.0 
14.8 
14.1 
12.9 

11.5 

9.6 

7-5 
5-2 
2.7 

o 
O.O 
O.O 
0.0 
O.O 
0.0 
O.O 
0.0 
0.0 
O.O 

0 

15-0 
14.8 
14.1 
12.9 

"•5 

9.6 

7-5 
5-2 
2.7 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name. 

o 

0 

10 

20 
30 
4O 

50 
I  60 
7O 
80 

0 

o 

10 
20 
30 
40 
50 
60 
70 

0 

O 
10 

20 
30 
40 
50 
60 

o 

O 
10 

20 

30 
40 
50 

0 

80.0 

70.0 
60.0 
50.0 
40.0 
30.0 

2O.O 
10.0 
O.O 

o 

72.0 

65.0 

56.5 

47-5 
38.1 
28.6 
18.9 
9-3 

O.I 

58^5 
54-1 
47-9 
40.7 
32.8 

24-5 
1  6.0 

74 

1.2 

O 
44.1 
41.0 
36.5 
3I-I 
2J.9 
18.3 
II.4 

4-3 

2.9 

0 

29.5 
27.0 

23-7 
19.8 

154 
10.5 

54 

°-3 

4.9 

14.8 

12.8 

10.4 

p 

2-3 

I.I 

4.4 

0 

0.0 

w 

3-4 
5-o 
6.4 
7.6 

o 

14.8 

o 

O 
10 
20 
30 
40 
5O 
60 
70 
80 

Pol.  Dist.  110°,  or  Declination  20°  of  contrary  name. 

0 

70.0 
60.0 
50.0 
40.0 
30.0 

2O.O 
1  0.0 
O.O 

o 

65.2 
56.5 

474 
38.0 
28.4 
18.8 
9.1 

O.2 

O 

54-5 
47-9 
40.4 
32.2 
23.8 

i5-i 

6.4 

2.4 

9. 

41.6 

3^5 
30-5 
23-9 

16.8 
94 

2.O 

5-4 

28°0 

23.7 
18.9 

13.6 

8.1 
2.4 

3-4 

0 

14.1 

10.4 

6-5 

2.4 
i.9 

6.1 

0 

0.0 

3-9 
6.7 

0 

14.1 

0 

O 
10 
20 
30 
40 
50 
60 
70 

Pol.  Dist.  120°,  or  Declination  30°  of  contrary  name. 

0 

0 
10 
20 
30 
40 
50 
60 

o 

0 
10 
20 
30 
40 
50 

,   o 

oo.o 
50.0 
40.0 

30.0 

20.0 
IO.O 
O.O 

56°8 

474 
38.0 

28.3 
18.6 
8.8 
0.5 

o 

48.6 
40.7 
32.2 

23-5 
14.6 

5.8 

3-2 

o 

37-8 
3i-i 
23-9 
16.2 

8.5 

0.9 

7-3 

o 
25.6 
19.8 
I3.6 

7-i 

0.4 

6.0 

o 

12.9 

7.7 
2.3 

3-2 

8.3 

o 
0.0 

5-i 
9-9 

o 
129 

Pol.  Dist.  130°,  or  Declination  40°  of  contrary  name. 

0 

50.0 
40.0 
30.0 

2O.O 
IO.O 
0.0 

0 

47-7 
38.1 
28.3 
18.7 
8.8 

0.6 

0 

41-5 
32.8 

23.8 

14.6 

5-5 

3-6 

o 
32.8 
25.0 

1  6.8 
8.6 
.    0.5 
8.3 

o 
22.5 

'fc 

0.4 

6.9 

0 

11.4 

4.8 

1.9 

8.2 

0 

0.0 

6.4 

0 

11.4 

138 


TABLE  XXXIV. 


Error  of  the  Time-Azimuth  for  an  Error  of   1     in  the  Hour-Angle. 

Hour-Angle  lh  or  llh. 

Az. 

Position-  Angle. 

Az. 

90° 

85° 

80° 

75° 

70° 

653 

60° 

5-5° 

50° 

45° 

40° 

30° 

20= 

10° 

0=> 

o 

O 
5 
10 
15 

20 
25 

30 
3.) 
40 
45 

M 
!  60 
|  70 
!  SO 
9O 

O7 
0 

o 
o 

0 

o 

0 
0 
0 
0 
0 

o 

0 
0 
0 

O7 
0 

I 
I 

2 
•2. 
3 

3 
3 
4 
4 
4 
5 
5 
5 

0' 

i 

2 

3 
3 
4 

I 

6 
7 
8 
9 
9 
10 

10 

o' 
I 

3 
4 
5 
6 
7 
9 

10 

ii 

12 
13 
14 
15 

J5 

o' 

2 

3 

5 
7 
8 

10 

ii 
13 
H 
'5 
17 
19 
20 

20 

o' 

2 

6 

8 

10 
12 
14 

16 
17 
19 

21 

23 
24 

25 

0' 

3 
5 
7 

10 
12 
H 
17 
19 
2O 

22 

25 

3 

29 

O7 

i 

9 
ii 

H 
17 
^9 

21 
23 
26 
29 
31 

33 
33 

0' 
10 

13 

16 
19 

21 

24 
26 
28 
32 

35 
37 
37 

o7 
4 
7 
ii 

14 

17 

20 

8 

29 
31 

p 

40 
41 

o' 

i 

12 
15 

19 
22 
26 
28 
31 

34 
39 
42 

44 
44 

0' 

4 
9 
13 
17 

21 

25 
29 
32 

35 
39 
43 
47 
49 
50 

0' 

5 
9 
H 
19 
23 
27 
31 

? 

42 
47 
5i 
54 
5,5 

o' 

5 

10 

15 

20 

11 

33 
37 
40 

44 
49 

? 

57 

O7 

5 

IO 

15 

20 

25 
29 

33 
37 
4i 
44 
5« 
55 

i 

1§0 
175 
17O  , 
165 
160 
155 
150  j 
145 
140 
135 
130 
12O 
110 
100 
90 

Az. 

90° 

95° 

100 

105°!  110° 

115° 

120° 

125° 

130°  1  135° 

140° 

150° 

160° 

170° 

180° 

Az. 

Position-  Angle. 

1  Az. 

Hour-Angle  2h  or  10h. 

Position-  Angle. 

Az. 

90° 

85° 

80° 

75° 

7O° 

65° 

0' 

I 

2 

3 

4 

6 

I 

9 

10 

ii 

12 
12 

y 

60° 

O7 

3 
4 
5 

6 

7 
9 

IO 

ii 

12 
'3 
14 
15 
15 

55° 

50° 

45° 

40° 

30° 

20° 

10° 

0? 

0 

0 
5 
10 
15 
20 
25 
30 
35 
40 
4> 

50 
60 
70 
80 
90 

• 
0' 
0 
0 

o 

0 
0 
0 

o 

0 

o 
o 

0 
0 

o 

0 

o' 
o 

0 

2 
2 
2 
2 

3 
3 
3 

0' 

o 

I 
I 

2 
2 
3 

3 
3 
4 
4 
5 
5 
5 
5 

o' 

I 

2 

3 
3 
4 
4 

I 

6 

7 

8 

0' 
2 

3  . 
3 
4 

7 
7 
8 

9 
10 

10 
IO 

o' 
i 
3 
4 
6 

7 
9 

10 

ii 

12 
*3 

!l 

17 
*7 

0' 
2 

3 

5 
7 
8 

10 

ii 

12 
14 
15 

19 
19 

o7 

2 

I 

7 

9 
ii 

12 

H 
15 

16 
18 

20 
21 
21 

0' 
2 

4 
6 
8 

IO 
12 
!3 
15 

16 
18 

20 
22 
23 
23 
140° 

o' 

2 

5 
7 
9 
ii 
13 
15 
17 
18 

20 

22 
24 
26 
2_6_ 
150° 

0' 

3 

5 
7 

IO 
12 
H 

16 
18 
20 

22 
24 
26 
28 
28 
160° 

o' 
3 

8 

IO 
12 
15 
17 
19 
21 

s 

28 
29 

29 

o' 

i 

IO 

13 

15 
•17 
19 

21 

23 
26 
28 
29 

>o 

1*0 
175  i 
170 
165 
160 
155 
150 
145 
140 
135 
13O 
120  i 
a  §  >  j 

1OO  i 

90  i 

Az. 

90° 

95°   1OO°  105° 

110 

115° 

120° 

125° 

130° 

135° 

170°  J  180° 

Az. 

Position-  Angle. 

Hour-  Angle  3U  or  9h. 

Az. 

Position-  Angle. 

Az.  | 

9<F    85o 

80° 

75° 

70° 

65° 

60° 

55° 

50° 

45° 

40o 

30- 

20°     1O3 

0° 

o 

0 
5 
10 
15 

20 
25 
30 
35 
40 
45 
50 
60 
70 
SO 
9O 

o' 
o 

0 
0 

o 
o 

0 
0 

o 
o 

0 
0 
0 

o 
o 

o' 
o 
o 

2 
2 
2 

2 

o' 
o 

I 
I 

2 
2 
2 

2 

3 

3 
3 
4 
4 
4 

o' 

i 
i 
i 

2 
2 
3 

3 
3 
4 
4 
5 

i 

o' 

I 
I 

2 
2 

3 
4 
4 
5 

6 
6 

7 
7 

7 

0'* 

2 
2 

3 
4 
4 

I 

6 

8 
9 
9 

O7 

I 

2 

3 
4 
4 

7 
7 
8 
9 

10 
10 

ii 

o' 

2 

3 

4 

I 

9 
9 

10 

ii 

12 
12 

0' 
2 

3 
5 
6 

9 

10 
10 
12 

13 
13 
14 

0 

3 
4 
5 
6 
7 
9 

10 

ii 

12 
13 
H 
15 
15 

0' 

3 

1 

8 
9 

10 
12 

13 

14 

8 

16 

0' 
2 

3 

8 
9 

IO 
12 
13 

\l 
\l 

18 

0' 

2 

^ 

8 

10 

ii 
13 
H 
15 
l? 
19 

20 
20 

o7 

2 

4 

5 
7 
9 

IO 
12 
13 
15 

16 
18 

20 
21 
21 

o' 

2 

4 
6 

7 

9 
ii 

12 
H 
15 

16 
18 

20 

21 
21 

180 
17  > 
170 
165 
160 
155 
150 
140 
145 
135 
13O 
120 
110 
1OO 
9O  i 

Az. 

90°    95° 

100° 

105° 

110° 

1  15°  j  120° 

125- 

130° 

135° 

14O° 

150° 

160°J17O° 

180° 

Azl 

Position-  Angle. 

TABLE  XXXIV. 


139 


Error  of  the  Time-Azimuth  for  an  Error  of  lm  in  the  Hour-Angle. 

Hour-  Angle  4h  or  8h. 

Az. 

Position-  Angle. 

Az. 

90° 

85° 

80° 

75° 

70° 

65° 

60° 

55° 

50° 

45° 

40° 

30° 

20° 

10° 

0° 

o 

5 
10 
15 
20 
25 
30 
35 
4O 
45 
50 
60 
70 
§0 
90 

Az. 

o1 
o 
o 
o 

0 
0 

o 
o 

0 

o 
o 
o 

0 

o 
o 

o' 

0 

o 
o 

I 

2 

o' 
o 
o 
I 
I 
I 
I 

2 
2 
2 
2 
3 

3 
3 
3 

o' 

0 

I 

I 
I 

2 
2 

3 
3 
3 
3 
4 
4 
4 
5 

o' 
I 
I 

I 

2 
2 

3 
3 
4 
4 
4 

6 
6 
6 

0' 

I 

I 

2 

2 

3 
4 
4 
5 
5 
6 
6 
7 
7 
7 

o' 

i 
i 

2 

3 

4 

4 

6 

7 

8 
9 

o' 

2 

3 
3 
4 

I 

6 
7 
8 
9 
9 
10 

IO 

o' 
I 

2 

3 
4 

I 

6 

8 
9 

10 
10 

ii 
ii 

o' 

i 

2 

3 
4 

i 

9 

9 
ii 
ii 

12 
12 

o' 

2 

3 
4 
6 

8 
9 
9 

10 

ii 

12 

'3 

J3 

0' 

i 
3 
4 
5 
6 
7 
9 

10 

ii 
ii 

13 
14 
15 
15 

o' 

i 
3 
4 
6 

9 

10 

ii 

12 

H 
15 
IO 

16 

o' 

I 
3 

I 
I 

10 

ii 

12 
13 

11 

17 
17 

o' 

2 

3 

7 
9 

10 

ii 

12 
13 

16 
17 

J7 

180 
175 
170 
165 
160 
155 
150 
145 
14O 
135 
130 
12O 
110 
100 
90 

90 

95° 

100°  105°  11O° 

115° 

12O° 

125° 

130° 

135° 

140° 

150° 

16O° 

17O° 

180° 

Az. 

Position-  Angle. 

Hour-Angle  5h  or  7h. 

Az. 

T 

5 
10 
15 

20 
25 
30 
35 
40 
45 
50 
60 
70 
§0 
90 

Position-  Angle. 

Az. 

90° 

85° 

80° 

75° 

70° 

65° 

60° 

55° 

50° 

45° 

4O° 

30° 

2<P 

10° 

0° 

o' 
o 

0 

o 
o 

0 

o 

0 

o 
o 

0 

o 

0 
0 

o 

o' 
o 
o 

0 

o 

I 

o' 
o 

0 

2 
2 
2 
2 
2 

3 
3 

o' 

0 

I 
I 

I 

2 
2 
2 
3 

3 

3 
3 
4 
4 
4 

o' 

0 

I 
I 

2 

2 

3 
3 
3 

4 

5 

5 
S 
S 

o' 
i 
I 

2 
2 

3 

3 
4 
4 

6 
6 
6 
7 

o' 

2 

3 
3 
4 
4 

5 
5 
6 
7 

8 
8 

0' 

I 

2 

3 
4 
4 

I 

6 

8 
8 
9 
9 

o' 
i 

2 

3 
3 
4 

i 

6 
7 
8 
9 
9 

10 
IO 

0' 

I 

2 

3 
4 

5 

8 
9 

IO 

ii 
ii 

0' 

I 

2 

3 
4 

8 
8 

9 

IO 

ii 

12 
12 

o' 
I 

2 

3 
5 
6 

1 

9 
9 

10 

ii 
13 
13 

El 

o' 
I 

2 

4 
5 
6 

9 

10 

ii 

!3 

H 
H 
'5 

o' 
I 

3 
4 
5 
6 
8 
9 

IO 

ii 

12 
13 
14 

15 

'5 

0' 

i 
3 
4 
5 

I 

9 

10 

ii 

12 
13 
15 

If 

180 
175 
170 
165 
160 
155 
15O 
145 
140 
135 
130 
120 
110 
100 
90 

Az. 

90° 

95°  1  1OO°  105° 

110°  1-15°  12O° 

125° 

13(P 

135° 

140°  150° 

160° 

170° 

180° 

Az. 

Position-  Angle. 

Hour-Angle  6h. 

Az. 

Position-  Angle. 

Az. 

90° 

85° 

80°      75° 

70° 

65° 

o' 
i 

2 

2 

3 
3 
4 
4 
4 

I 

6 
6 

60° 

55° 

50°     45°  !  40° 

30° 

20° 

10° 

0° 

O 
5 
1O 
15 
20 
25 
3O 
35 
40 
45 
50 
6O 
70 
80 
9O 

Az. 

0' 

o 
o 

0 

o 
o 

0 

o 
o 

0 
0 

o 
o 

0 

o 

o' 

0 

o 
o 

0 

o' 
o 
o 

2 
2 
2 
2 
2 

3 
3 

o' 
o 
i 
i 

i 

2 
2 
2 
2 

3 
3 
3 
4 
4 
4 

o' 

O 

I 

2 
2 

3 
3 
3 
4 
4 
4 
5 
5 
5 

o' 
I 
i 

2 

3 
3 
4 
4 
5 
5 
6 
6 
7 
7 
7 

o' 
I 

I 

2 

3 

4 
4 
5 

7 

I 

8 
9 

o' 

i 

2 
2 
3 

4 

5 

7 

9 
9 

10 

o' 

i 

2 

3 
4 
4 

I 

7 
7 
8 
9 

10 
10 

ii 

o' 

2 

3 
4 

I 

7 
9 

10 

n 
ii 

12 

o' 

i 

2 

3 
4 

i 

9 

10 

ii 

12 

'3 
J3 

o' 

I 

2 

4 
5 
6 

8 
9 

10 

ii 

12 
13 
H 
H 

o' 
i 
3 
4 
5 
6 

8 
9 

10 

ii 
'3 
14 
15 
15 

0' 

3 
4 
5 
6 
7 
9 

10 

ii 

12 
13 
14 
15 
15 

180 
173 
170 
165 
160 
155 
150 
145 
140 
135 
13O 
120 
110 
10O 
90 

9(P 

95°   1OO° 

105°  110° 

115°  120° 

125° 

130° 

1353 

140° 

150° 

160° 

170° 

180° 

Az. 

Position-  Angle. 

140 


TABLE  XXXIV. 


Error  of  the  Time-Azimuth  for  an  Error  of  I     in  the  Hour-  Angle. 

Hour-Angle  Oh  10m  or  llh  50m. 

Az. 

Position-  Angle. 

Az. 

90° 

85° 

80° 

75° 

70° 

65° 

60° 

55° 

50° 

45° 

40° 

30° 

20° 

10° 

0° 

0 
5 
10 
15 
2O 
25 
30 
3-> 
40 
45 
50 
60 
70 
80 
i  90 

0 

O.O 

o 

0 
0 

o 
o.o 
o 

0 

o 
o 
o.o 
o 

0 

o 
o 

o 
0.0 

o 
I 

I 

2 
0.2 
2 

3 

3 
4 
0.4 

4 
5 

5 

g 

o 
0.0 

I 

t 

3 
3 
0.4 

6 
6 
7 
0.8 
9 
9 

I.O 

o 

0 

0.0 

I 
3 
4 
5 
0.6 
7 
9 

I.O 
0 

i.i 

3 
4 
5 
5 

0 

O.O 

2 

3 
5 
7 
0.8 

I.O 

i 

3 
4 

i-5 

I 

9 

2.0 

o 
0.0 
2 

4 
6 
8 

I.O 
2 

I 

7 
1.9 

2.1 

3 
4 
4 

0 

O.O 
2 

S 
7 

I.O 
1.2 

4 
6 
8 

2.0 
2.2 
5 

8 
9 

o 
O.O 

I 

9 
i.i 

1.4 
6 
9 

2.1 

3 
2.5 

3-i 

2 

3 

o 
O.O 

3 
6 

I.O 

3 

1.6 
8 

2.1 

4 
6 

2.8 

3-2 

I 

7 

o 
0.0 

4 
7 

I.O 

4 
i-7 

2.0 

9 
3-i 

4.0 

o 
0.0 

\ 

I.I 
5 
1.9 

2.2 

3-1 

fj 

4-1 

3 
4 

o 
O.O 

4 
9 
i-3 

7 

2.1 

§ 

3-2 
5 
3-8 
4-3 
7 
9 
5-o 

o 
0.0 

5 
9 
1.4 
8 

2-3 
7 
3-i 

4.1 
7 
5-i 
3 
4 

o 
0.0 

5 

I.O 

5 
9 
2.4 
8 

36 

4.0 

4-3 
9 

5i 

6 

o 
0.0 

5 

I.O 

5 

2.O 
2.4 

9 
3-3 
7 
4.1 

4-4 
5-o 
4 
6 

7 

180 
175 
170 
165 
160 
155 
150 
145 
140 
135 
13O 
120 
110 
1OO 
90 

Az. 

90° 

95° 

100° 

105°  110° 

115° 

120° 

125° 

130° 

135°  140° 

150° 

160° 

170°  180° 

Az. 

Position-  Angle. 

Hour-Angle  Oh  20m  or  llh  40m. 

Az. 

Position-  Angle. 

Az. 

90° 

85° 

80° 

75° 

70° 

65° 

60° 

55° 

50° 

45° 

40° 

30° 

20° 

10° 

O3 

0 

O 
5 
10 
15 
20 
25 
30 
35 
40 
43 
50 
60 
7O 
80 
90 

0 

0.0 

o 

0 

o 
o 
o.o 
o 

0 
0 

o 

o.o 

0 
0 

o 
o 

90° 

0 

0.0 
0 

o 

0. 

2 
2 
O.2 
2 
2 
2 

3 

0 

O.O 

o 
I 

I 

2 
0.2 
2 
3 

3 
4 
0.4 
4 
5 
5 
f 

0 

0.0 

I 

I 

2 

3 

0-3 
4 
4 
5 
5 
0.6 
6 
7 
7 
7 

o 
0.0 
I 
2 

3 

3 
0.4 

I 

6 
7 
0.8 
9 
9 

I.O 
0 

o 
0.0 

I 

2 

3 
4 

°-5 
6 

8 
9 
0.9 
i.i 
i 

2 
2 

0 

0.0 
I 

2 

4 
5 
0.6 

8 
9 

I.O 

i.i 

2 

3 

4 
4 

0 

O.O 

I 

3 

4 
6 

0.7 
8 
9 

2 

i-3 

4 

6 
6 

o 
O.O 

2 

I 

0.8 
9 

2 

3 

'1 

8 
8 

o 

o.o 

2 

4 
5 
7 
0.9 

I.O 

2 

3 
4 
1.6 
8 
9 

2.0 

o 

o 
O.O 
2 

4 
6 
8 
0.9 
i.i 
3 

i 

i-7 

9 

2.1 
2 
2 

o 
0.0 
2 

| 

9 
i.i 

2 

i 

8 
1.9 

2.2 
3 

4 
150° 

o 
O.O 

2 

S 
7 
9 
i.i 
3 
5 
7 
9 

2.1 

3 
S 
7 
7 

o 
O.O 
2 

5 
7 

I.O 
1.2 

4 
6 
8 

2.0 
2.2 

4 

I 

8 

o 

o.o 
3 

5 
7 

I.O 
1.2 

4 
6 
8 

2.0 
2.2 

I 

9 

180 
175 
170 
165 
160 
155 
15O 
145 
140 
135 
130 
12O 
110  , 
1OO 
90 

Az. 

i 

95° 

100° 

105°  j  110=> 

115° 

120° 

125° 

130° 

135° 

140? 

160° 

170° 

180° 

Az. 

Position-  Angle. 

Hour-Angle  Oh  30ra  or  llh  30'".                                                      j 

Az. 

Position-  Angle. 

Az. 

90°    85° 

80° 

75° 

70° 

65° 

60° 

55° 

50° 

45° 

40° 

30° 

20° 

10° 

0° 

0 
5 
1O 
15 
20 
25 
30 
35 
40 
45 
50 
60 
70 
80 
90 

0 

0.0 
0 
0 

o 

0 
0.0 
0 

o 
o 

o 
o.o 

0 
0 
0 

o 

o 
0.0 
0 

o 
o 
I 

O.I 

I 

I 
I 
I 

O.I 

I 

2 
2 
2 

o 
O.O 
0 
I 
I 
I 
O.I 
2 
2 
2 
2 

o-3 
3 
3 
3 

3 

o 
0.0 
0 

I 

I 

2 
0.2 
2 
3 

3 
3 
0.4 
4 
5 
5 
5 

o 
0.0 

I 

I 

2 
2 

0-3 

3 

4 
4 
5 

1 

6 
6 

7 

0 

0.0 

I 
I 
2 

3 

0.3 

4 

5 

I 

0.6 

1 

8 
8 

0 

0.0 
I 

2 
2 
3 
0.4 

i 

7 

1 

9 
9 

I.O 

0 

0.6 
I 

2 

3 
4 

°-5 

I 
I 

0.8 
9 

I.O 

I 

o 

o.o 

I 

2 

3 
4 
0.5 

I 

9 
0.9 

I.O 

2 
2 
2 

o 
O.O 

I 

2 

3 

5 
0.6 

8 
9 

I.O 
I.O 
2 

3 
3 
4 

o 
0.0 

I 

3 
4 
5 
0.6 

I 

9 

I.O 

3 

4 
4 

5 

0 

0.0 

I 
3 

6 
0.7 
8 
9 

I.O 
2 

i-3 
4 
6 
6 

7 

0 

0.0 

2 

i 

0.8 
9 

I.O 
2 

3 

'1 

87 
8 

0 

O.O 

2 

3 

1 

0.8 

9 
i.i 

2 

3 

1.4 

6 
8 
9 
9 

o 
0.0 
2 

3 
5 
7 
0.8 

I.O 

i 

2 

4 

1 

9 
9 

ISO 
175 
170 
165  i 
160 
155 
15O 
145 
140 
135 
13O 
12O 
110 
100 
9O 

u*. 

90° 

93° 

100° 

105° 

110° 

115° 

l*O° 

125° 

130° 

135^ 

140° 

150° 

160° 

170° 

18O° 

Az. 

Position-Angle. 

TABLE  XXXV. 


141 


Error  of  Time-Azimuth  for  an  Error  of  12  ,  or  O°.2,  in  Latitude. 

Altitude. 

A  »* 

Az. 

JU. 

O° 

10° 

20° 

30° 

40°  i  50° 

60° 

7O° 

75° 

8O° 

82° 

84° 

86° 

88° 

90 

o 

o 

o' 

o' 

o' 

o' 

o' 

0' 

o' 

o' 

o 
0.0 

0 

0.0 

0 

O.O 

o 
O.O 

o 
O.O 

o    . 
0.0 

bidet. 

ISO 

2 

o 

o 

o 

o 

o 

i 

i 

i 

O 

0 

I 

I 

I 

2 

00 

178 

4 

o 

o 

o 

I 

i 

i 

2 

I 

I 

I 

I 

2 

4 

00 

176 

6 

o 

0 

1 

2 

2 

3 

I 

I 

2 

2 

3 

6 

00 

174 

8 

0 

o 

I 

2 

3 

5 

I 

2 

2 

3 

4 

8 

00 

172 

1O 

o 

0 

2 

3 

4 

6 

O.I 

O.2 

o-3 

0.3 

0.5 

I.O 

00 

170 

12 
14 

o 
o 

o 

2 

2 

2     I 

3 
4 

4 
5 

I 

2 
2 

2 

3 

3 
3 

4 
5 

6 
7 

2 

4 

00 
OO 

168 

166 

16 

o 

2 

3 

4 

6 

9 

2 

3 

4 

5 

8 

6 

00 

164 

18 

o 

2 

3 

4 

6 

10 

2 

3 

4 

6 

9 

8 

00 

162 

2O 

o 

2 

2 

3 

5 

•  7 

ii 

°'3 

0.4 

0.5 

0.7 

I.O 

2.0 

00 

160  ! 

25 

o 

2 

3 

4 

6 

8 

14 

3 

5 

6 

8 

2 

4 

00 

155 

3O 

o 

2 

7 

n 

10 

16 

4 

6 

7 

I.O 

4 

9 

OO 

150 

35 

o 

3 

O 

4 

6 

8 

12 

19 

4 

6 

8 

i 

6 

3-3 

00 

145  1 

4O 

0 

3 

5 

7 

9 

'3 

21 

5 

7 

9 

2 

8 

7 

00 

140 

45 

0 

2. 

3 

5 

10 

15 

23 

0.5 

0.8 

I.O 

1.4 

2.0 

4.1 

00 

135 

5O 

o 

2. 

3 

8 

ii 

16 

25 

6 

9 

i 

5 

2 

4 

00 

130 

55 

o 

2. 

4 

6 

8 

12 

!7 

27 

6 

9 

2 

6 

3 

7 

00 

125 

60 

o 

2. 

4 

6 

9 

12 

18 

29 

6 

I.O 

2 

7 

5-° 

00 

12O 

65 

0 

2. 

4 

6 

9 

13 

19 

3° 

7 

0 

3 

7 

6 

2 

00 

115 

70 

o 

2. 

4 

7 

10 

'3 

19 

0.7 

i. 

1.3 

1.8 

2-7 

54 

00 

110 

75 

0 

2, 

4 

7 

10 

H 

20 

32 

7 

4 

8 

8 

5 

00 

1O5 

80 

o 

2. 

4 

7 

10 

H 

20 

32 

7 

4 

9 

8 

6 

00 

1OO 

85. 

o 

2. 

4 

7 

10 

H 

21 

33 

7 

4 

9 

8 

7 

00 

95 

90 

0 

2 

4 

7 

10 

H 

21 

33 

7 

4 

9 

9 

7 

00 

9O 

•                                                                                                                                            i 

TABLE  XXXVI. 

Error  of  Time-Azimuth  for  an  Error  of  6',  or  O°.l,  in  Declination. 

Pos. 

Altitude. 

Pos. 

Ang. 

0° 

10° 

2O° 

30° 

4O° 

50° 

6O° 

70° 

75° 

80° 

82° 

84° 

86" 

88> 

90° 

Ang. 

o 

0 

0' 

o' 

o' 

o' 

o' 

o' 

o' 

o' 

0 

O.O 

0 

0.0 

o 
O.O 

0 

0.0 

0 

0.0 

o 
0.0 

Indet. 

180 

2 

0 

0 

o 

o 

0 

0 

o 

i 

0 

0 

O 

o 

I 

I 

00 

178 

4 

o 

o 

0 

I 

I 

I 

i 

o 

o 

I 

I 

I 

2 

00 

176 

6 

I 

I 

I 

I 

I 

2 

0 

I 

I 

I 

2 

3 

00 

174 

8 

I 

I 

I 

I 

2 

2 

I 

I 

I 

I 

2 

4 

OO 

172 

10 

I 

I 

I 

2. 

2 

3 

O.I 

O.I 

O.I 

0.2 

°-3 

0.5 

00 

170 

i   12 

I 

I 

2 

2. 

3 

4 

I 

I 

2 

2 

3 

6 

00 

168 

14 

2 

2 

2 

2 

2 

3 

4 

I 

I 

2 

2 

3 

7 

00 

166 

16 

2, 

2 

2 

2 

2 

3 

3 

5 

I 

2 

2 

3 

4 

8 

00 

164 

18 

2, 

2 

2 

2 

2 

3 

4 

5 

I 

2 

2 

3 

4 

9 

00 

162 

20 

2. 

2 

2 

2 

3 

3 

4 

6 

O.I 

0.2 

•0.2 

0-3 

0-5 

I.O 

00 

16O 

25 

2, 

3 

3 

3 

3 

4 

5 

7 

2 

2 

3 

4 

6 

2 

00 

155 

30 

3 

3 

3 

4 

4 

5 

6 

9 

2 

3 

4 

5 

7 

4 

00 

150 

35 

3 

4 

4 

4 

5 

7 

10 

2 

3 

4 

5 

8 

6 

00 

145 

40 

4 

4 

4 

5 

-5 

6 

8 

ii 

2 

4 

5 

6 

9 

8 

00 

140 

45 

4 

4 

5 

5 

6 

7 

9 

12 

0.3 

0.4 

0-5 

0.7 

I.O 

2.0 

00 

135 

5O 
55 

5 
5 

5 
5 

5 
5 

6 
6 

8 

9 

10 

'3 

14 

3 
3 

4 
5 

I 

I 

i 

2 

2 

4 

00 
00 

130 
125 

60 

5 

6 

6 

7 

8 

10 

3 

5 

6 

8 

2 

00 

120 

65 

5 

6 

6 

6 

7 

9 

ii 

16 

3 

5 

7 

9 

3 

6 

00 

115 

70 

6 

6 

6 

7 

7 

9 

ii 

17 

0.4 

0.5 

0.7 

0.9 

1.3 

2.7 

00 

110 

75 

6 

6 

6 

7 

8 

9 

12 

17 

4 

6 

7 

9 

4 

8 

00 

105 

80 

6 

6 

6 

7 

8 

9 

12 

17 

4 

6 

7 

9 

4 

8 

OO 

1OO 

85 

-6 

6. 

6 

7 

8 

9 

12 

18 

4 

6 

7 

I.O 

4 

9 

00 

95 

9O 

6 

6 

6 

7 

8 

9 

12 

18 

4 

6 

7 

o 

4 

9 

00 

9O 

142 


TABLE  XXXVII. 


Limiting*  Errors  of  Time-Azimuths. 

Hour- 
Angle. 

First  Supposition. 

Second  Supposition. 

Third  Supposition. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

Partial  Az.  Error. 

rrOD. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

±1.9  | 

I.O 

0.8 
0.8  ; 
0.8 
0.8  i 
0.7 
0.7 
0.7 
0.7 

H.A. 

Error 

±  lm. 

Lat. 

Error 
±  12'. 

o     | 

o's 
0.4 

O.2 
O.I 
O.I 
O.I 
O.O 

o.o 
o.o 

0.0 

Dec. 
Error 
±3'. 

H.  A.       Lat. 
Error     Error 

±  2».       ±  IS'. 

n           Total 

AZ. 

Y5?p    Error. 

H.A.   j    Lat. 
Error     Error 
±  3-».      ±  30'. 

Dec.    i 
Error 
±3. 

h     m 
1     0 
2    0 
3    0 
4    0 
5    O 
6    0 
7    O 
8    0 
9    0 
i   1O    O 
i  11     0 
12    0 

o 

±0.6 
0.4 
o-3 

o-3 
o-3 

o-3 
0-3 

0 

±0.2 
0.2 
O.I 
O.I 
O.I 
O.I 
O.O 
0.0 
0.0 

o.o 
o.o 

0.0 

0 

±1.0 
0.7 

0.6 

0-5 
0.4 
0.4 
0.4 

o-3 

o-3 
0-3 

o 

±0.9 
0.7 

o-5 
0-5 
o-5 
0-5 
0-5 

0-5 
0.5 

o-5 

0 

0.8 

o-5 
0.4 

0-3 

0.2 
O.I 
O.I 
O.I 
0.0 

o.o 
o.o 

o 
±0.2 
0.2 
O.I 
O.I 
O.I 
O.I 
O.O 

e.o 
o.o 

0.0 
0.0 
0.0 

o 
±1.4 
I.I 

0.8 
0.7 
0.6 
0.6 
0.6 
0.6 
0-5 
o-5 

o-5 

o 

±1.9 

I.I      : 

0.9     i 

0.8   ; 

0.7 
0.7 

0.7: 

0.7 
0.7 

0.7 
0.7 
0.7  i 

o 
±1.9 

0.9 

0.6 
0.4 
o-3 

O.2 
0.2 
O.I 
O.I 
0.0 
0.0 

o 
±0.1 
0.2 
O.I 
O.I 
O.I 
O.I 
O.O 

o.o 

0.0 

o.o  ! 
o.o 

0.0 

TABLE  XXXVIII. 

Limiting:  Errors  of  Ti  sae-Azimuths  in  High  Latitudes. 

Hour- 
Angle. 

First  Supposition. 

Second  Supposition. 

Third  Supposition. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

±1-. 

±  12'. 

±3'. 

±2-. 

±18'. 

±3'. 

±3*. 

±30'. 

±0'. 

h     m 
0    0 
010 
O3O 
1    0 
2    O 
3    0 
4    0 
5    0 
6    0 
7    0 
8    0 
9    0 
1O    O 
11    0 
12    O 

o 

±0.4 
0.4 
0.4 
0.4 

o-3 

o-3 
0-3 
0-3 
o-3 
o-3 
o-3 

0 

0.0 
0.0 
0.0 
±0.1 
0.2 
O.2 
0.2 
O.I 
O.I 
O.I 
O.I 
0.0 
O.O 

o.o 
o.o 

o 

o.o 

0.0 

o.o 

0.0 
0.0 
0.0 
0.0 

o.o 

0.0 
0.0 
0.0 
0.0 

o.o 

0.0 
0.0 

0 

±0.4 
0.4 
0.4 
0.4 
0.4 
0.4 
0.4 
0.4 
0.4 

o-3 
0-3 

0-3 

0 

±0.7 
0.7 
0.7 
0.7 
0.6 

0-5 
0-5 
o-5 

o-5 
0-5 

0-5 

0 

0.0 

o.o 

±0.1 
0.2 
0.2 
0.2 
O.2 
0.2 
0.2 
O.I 
O.I 
O.I 
0.0 
0.0 
O.O 

0 

o.o 

0.0 
0.0 
0.0 
0.0 
0.0 
0.0 

o.o 

0.0 
0.0 
0.0 
0.0 

o.o 
o.o 

0.0 

o 

±0.7 
0.7 
0.7 

0.8 
0.7 
0.6 
0.6 
0.6 
0.6 
o-S 
0-5 
o-5 

o-5 

o 

±1.1 
I.I 
I.I 
I.I 

0.9 

0.8 
0.8 
0.8 
0.8 
0.7 
0.7 
0.7 
0.7 
0.7 
0.7 

o 
0.0 
±0.1 
0.2 

0.4 
0.4 
0.4 
0.3 

O.2 
O.2 
O.I 
0.0 
O.O 
O.O 

o 

o.o 
o.o 
o.o 

±0.1 
O.I 
O.I 
O.I 
O.I 
O.I 
O.I 
O.I 
O.I 

o.o 

0.0 

o.o 

0 

±1.1 

I.I 
I.I 

1.2 
I.O 

0.9 
0.9 
0.9 

0.8 
0.8 
0.8 
0.7 
0.7 
0.7 
0.7 

TABLE  XXXIX. 

Limiting  Errors  of  Serial  Time-Azimuths. 

Hour- 
Angle. 

Observations  in  Port. 

At  Sea:  In  all  Latitudes. 

At  Sea:  In  all  Latitudes. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

Partial  Az.  Error. 

Prob. 
Total 
Az. 
Error. 

H.A. 
Error 
±4*. 

Lat. 
Error 

±1'. 

Dec. 
Error 
±  0'.5. 

H.A. 

Error 
±10-. 

Lat. 

Error 

±3'. 

Dec. 

Error 

±1'. 

H.A. 

Error 
±30». 

Lat. 
Error 
±0'. 

Dec. 
Error 

±1'. 

/'      m 
6    0 
5    0 
4    0 
3    0 
230 

±0.9 
0.9 

I.O 

1.3 

±0.4 
0.6 
0.8 

1.2 

1.6 

±0.5 

0.6 
0.8 

I.O 

±1.1 

1.2 

1.4 
1.9 
2.4 

±2.2 
2.2 
2.6 

3-3 
4.0 

±1.2 

4-9 

±1.1 
i.i 

1.2 

1.6 

1.9 

±2.7 

2.8 

3-7 

/ 

6.7 

7.8 

1  0.0 
I2.I 

±2.5 

34 
4.6 
7.2 
9-7 

±1.1 
i.i 

1.2 

1.6 

1.9 

±7.; 
7.6 

9.1 
15.6 

TABLE  XL. 


143 


4  i  rr  ii  111  polar  Azimuths:  Polaris,  or  the  Pole-Star. 

Western  Elongations. 

Sid.T. 

TUT          1 

Eastern  Elongations. 

Lat. 

Greatest 
Elongation 
West- 

Star  moving  fro  n  Meridian. 

Sid.  Time  of  Azimuths- 

Mer'n 
above 

Pole. 

Star  moving  towards  Merid  an. 

Sid.  Time  of  Azimuths- 

Greatest 
Elongation 
East- 

Lat. 

Sid.  T. 

Az. 

h  m 

613 

5  13 

413    313 

213 

113 

013 

2313 

2213 

21132013 

Az. 

Sid.  T. 

o 

h     m 

o 

0 

o 

o 

0 

o 

o 

0 

o 

0 

o 

o 

0 

h     m 

0 

O 

7  13-° 

1.4 

1.3 

1.2 

I.O 

0.7 

0-3 

0.0 

o.3 

0.7 

I.O 

1.2 

1.3 

1.4 

19  13.0 

O 

2 

12.8 

4 

3 

2 

O 

7 

3 

0 

3 

7 

O 

2 

3 

4 

13.2 

2 

4 

12.6 

4 

3 

2 

o 

7 

3 

o 

3 

7 

0 

2 

3 

4 

13-4 

4 

6 

12.4 

4 

3 

2 

o 

7 

4 

o 

4 

7 

o 

2 

3 

4 

13.6 

6 

8 

12.2 

4 

3 

2 

o 

7 

4 

o 

4 

7 

0 

2 

3 

4 

13.8 

8 

!  10 

7  12.0 

1.4 

1.3 

1.2 

I.O 

0.7 

0.4 

o.o 

0.4 

0.7 

I.O 

1.2 

1.3 

1.4 

19  14.0 

10 

12 

11.8 

4 

3 

2 

o 

7 

4 

0 

4 

7 

0 

2 

3 

4 

14.2 

12 

14 

11.6 

4 

4 

2 

o 

7 

4 

o 

4 

7 

o 

2 

4 

4 

14.4 

14 

16 

11.4 

4 

4 

3 

o 

7 

4 

o 

4 

7 

0 

3 

4 

4 

14.6 

16 

18 

11.2 

4 

4 

3 

o 

7 

4 

o 

4 

7 

o 

3 

4 

4 

14.8 

18 

20 

7  i  i.o 

1.5 

1.4 

1.3 

I.O 

0.7 

0.4 

0.0 

0.4 

0.7 

I.O 

1.3 

1.4 

1.5 

19  15.0 

20 

21 

10.9 

5 

4 

3 

0 

7 

4 

o 

4 

7 

0 

3 

4 

5 

21 

22 

10.8 

5 

4 

3 

o 

7 

4 

o 

4 

7 

o 

3 

4 

5 

15.2 

22 

23 

10.7 

5 

4 

3 

o 

7 

4 

o 

4 

7 

o 

3 

4 

5 

'5-3 

23 

24 

10.6 

5 

4 

3 

I 

7 

4 

o 

4 

7 

3 

4 

5 

15-4 

24 

25 

7  IO-5 

1.5 

1.4 

1.3     i. 

0.8 

0.4 

o.o 

0.4 

0.8 

I. 

J-3 

1.4 

1.5 

19  15-5 

25 

26 

10.4 

5 

5 

3 

8 

4 

o 

4 

8 

3 

5 

5 

15.6 

26 

27 

IO.2 

5 

5 

3 

8 

4 

o 

4 

8 

3 

5 

5 

15.8 

27 

28 

IO.I 

5 

3  j 

8 

4 

o 

4 

8 

3 

5 

5 

15-9 

28 

29 

IO.O 

6 

5 

4 

8 

4 

0 

4 

8 

4 

5 

6 

16.0 

29 

30 

7    9-9 

1.6 

1.5 

1.4     i. 

0.8 

04 

o.o 

0.4 

0.8 

I. 

1.4 

1.5 

1.6 

19  16.1 

30 

31 

9-7 

6 

5 

4 

8 

4 

0 

4 

8 

4 

5 

6 

16.3 

31 

32 

9.6 

6 

6 

4 

8 

4 

0 

4 

8 

I 

4 

6 

6 

16.4 

32 

33 

9-5 

6 

6 

4 

2 

8 

4 

o 

4 

8 

2 

4 

6 

6 

16.5 

33 

34 

9-3 

6 

6 

4 

2 

8 

4 

0 

4 

8 

2 

4 

6 

6 

16.7 

34 

35 

7    9-2 

1-7 

1.6 

1.4 

1.2 

0.8 

0.4 

o.o 

0.4 

0.8 

1.2 

1.4 

1.6 

l.y 

19  16.8 

35 

36 

Q.  I 

7 

6 

5 

2 

8 

4 

0 

4 

8 

2 

5 

6 

7 

16.9 

36 

37 

o.Q 

7 

6 

5 

2 

8 

4 

o 

4 

8 

2 

5 

6 

7 

17.1 

37 

38 

8.8 

7 

7 

5 

2 

9 

4 

o 

4 

9 

2 

5 

7 

7 

17.2 

38 

39 

8.6 

8 

7 

5 

2 

9 

5 

o 

5 

9 

2 

5 

7 

8 

17.4 

39 

40 

7    8.4 

1.8 

1.7 

'•5 

1.3 

0.9 

0.5 

o.o 

0.9 

1.3 

1.5 

1.7 

1.8 

19  17.6 

4O 

41 

8-3 

8 

7 

6 

3 

9 

5 

0 

5 

9 

3 

6 

7 

8 

17.7 

41 

42 

8.1 

8 

8 

6 

3 

9 

5 

o 

5 

9 

3 

6 

8 

8 

17.9 

42 

43 

7-9 

9 

8 

6 

3 

9 

5 

o 

5 

9 

3 

6 

8 

9 

18.1 

43 

44 

7.8 

9 

8 

6 

3 

I.O 

5 

0 

5 

I.O 

3 

6 

8 

9 

18.2 

44 

45 

7    7-6 

1.9 

1.9 

1-7 

1.4 

I.O 

o-5 

0.0 

I.O 

1.4 

1.7 

1.9 

1.9 

19  18.4 

45 

46 

7-4 

9 

9 

7 

4 

0 

5 

o 

5 

0 

4 

7 

9 

9 

1  8.6 

46 

47 

7.2 

2.0 

9 

4 

o 

5 

o 

5 

o 

4 

7 

9 

2.O 

18.8 

47 

48 

7.0 

o 

2.0 

8 

4 

o 

5 

o 

5 

0 

4 

8 

2.0 

O 

19.0 

48 

49 

6.8 

I 

0 

8 

5 

0 

5 

o 

5 

0 

5 

8 

O 

I 

19.2 

49 

50 

7    6.5 

2.1 

2.0 

1.8 

i.i 

0.5 

o.o 

I.I 

1.8 

2.0 

2.1 

19  J9-S 

50 

51 

6-3 

2 

I 

9 

5 

i 

6 

0 

6 

I 

5 

9 

I 

2 

19.7 

51 

52 

6.0 

2 

I 

9 

6 

i 

6 

0 

6 

I 

6 

9 

I 

2 

20.0 

52 

53 

5.8 

3 

2 

2.0 

6 

i 

6 

o 

6 

I 

6 

2.0 

2 

3 

20.2 

53 

54 

5-5 

3 

2 

0 

6 

2 

6 

0 

6 

2 

6 

0 

2 

3 

20.5 

54 

55 

7    5-2 

2.4 

2-3 

2.1 

1.7 

1.2 

0.6 

0.0 

0.6 

1.2 

1-7 

2.1 

2-3 

2.4 

19  20.8 

55 

56 

4-9 

4 

4 

I 

7 

2 

6 

o 

6 

2 

I 

4 

4 

21.  1 

56 

57 

4.6 

4 

2 

8 

3 

6 

o 

6 

3 

8 

2 

4 

5 

21.4 

57 

58 

4-3 

6 

5 

2 

8 

3 

7 

0 

7 

3 

8 

2 

6 

21.7 

58 

59 

4.0 

6 

6 

3 

9 

3 

7 

0 

7 

3 

9 

3 

6 

6 

22.O 

59 

6O 

7    3-6 

2.7 

2.6 

2.4 

2.0 

1.4 

0.7 

0.0 

0.7 

1.4 

2.0 

2.4 

2.6 

2.7 

19   22.4 

60 

Sid.  T. 

Az. 

111 

913 

1013 

1113 

1213 

h     m 

1313 

h  m 

1413 

1513 

1613 

1713 

1813 

Az. 

Sid.  T. 

Lat. 

Greatest 
Elongation 
West- 

Sid-  Time  of  Azimuths- 

Star  moving  towards  Meridian, 

Sid.T. 

M°e?'n 

below 
the 

Sid.  Time  of  Azimuths. 

Star  nwing  from  Meridian. 

Greatest 
Elongation 
East. 

Lat 

Pole- 

Subtract   R.  A.  True  Sun  (Tab.  LIX)  from  Sid.  T.  to  get  Ship  A.  T.,  or  subtract 

R.  A.  Mean  Sun  (by  applying  E.  T.  to  R.  A.  True  Sun)  to  get  Ship  M.  T. 

144 


TABLE  XLI. 


Altitude-  Azimuths  :  Part  1. 

Alt. 

Latitude. 

Alt. 

i 

°D 

2° 

4° 

6 

8= 

10 

12° 

14° 

16° 

iw 

20 

22 

24° 

26 

28° 

3O 

o 

0 

0 

0 

0 

i 

2 

3 

5 

6 

8 

ii 

13 

16 

20 

23 

27 

3* 

O  i 

2 

0 

0 

o 

i 

2 

3 

5 

6 

8 

ii 

13 

16 

20 

23 

27 

2 

4 

o 

o 

I 

2 

3 

4 

7 

9 

n 

17 

2O 

24 

28 

32 

4 

6 

I 

I 

2 

2 

3 

4 

6 

8 

10 

12 

15 

18 

21 

24 

28 

32 

6 

8 

2 

2 

3 

3 

4 

5 

7 

9 

n 

'3 

16 

19 

22 

25 

29 

33 

8 

10 

3 

3 

4 

4 

5 

6 

8 

10 

12 

H 

*7 

20 

23 

26 

3° 

34 

10 

12 

5 

5 

5 

6 

7 

8 

10 

ii 

13 

16 

18 

21 

24 

28 

32 

36 

12 

14 

6 

6 

7 

8 

9 

10 

ii 

13 

15 

17 

20 

23 

26 

30 

34 

38 

14 

16 

8 

8 

9 

10 

n 

12 

13 

15 

17 

19 

22 

25 

28 

32 

36 

40 

16 

18 

ii 

n 

ii 

12 

13 

H 

16 

17 

19 

22 

24 

27 

30 

34 

38 

42 

18 

20 

13 

13 

14 

15 

16 

17 

18 

20 

22 

24 

27 

30 

33 

37 

41 

45 

20 

22 
24 

16 

20 

16 
20 

17 

20 

21 

19 

22 

2O 
23 

21 

24 

2 

11 

27 

3° 

30 

33 

36 
39 

39 
43 

43 
47 

48 

22 
21 

26 

23 

23 

23 

24 

25 

26 

28 

3° 

32 

37 

40 

43 

46 

5° 

54 

26 

28 

27 

27 

27 

28 

29 

30 

32 

33 

35 

38 

4i 

44 

47 

50 

54 

58 

28 

30 

31 

3i 

31 

32 

33 

34 

36 

38 

40 

42 

45 

48 

51 

54 

58 

62 

30  i 

32 

36 

36 

36 

37 

38 

39 

41 

42 

44 

47 

49 

52 

59 

63 

67 

32 

34 

42 

43 

44 

46 

47 

49 

52 

54 

57 

60 

64 

68 

72 

34 

36 

46 

46 

46 

47 

48 

49 

51 

59 

62 

66 

69 

73 

77 

36 

38 

52 

52 

52 

53 

54 

55 

57 

58 

60 

63 

65 

68 

7i 

75 

79 

83 

38  | 

40 

58 

58 

S8 

59 

60 

61 

63 

64 

66 

69 

7i 

74 

77 

81 

85 

89 

40 

42 

64 

64 

65 

66 

67 

68 

69 

7i 

73 

75 

78 

81 

84 

88 

92 

96 

42 

44 

71 

72 

73 

74 

75 

76 

78 

80 

82 

85 

88 

•  91 

95 

99 

I03 

41  i 

46 

79 

79 

80 

80 

81 

82 

84 

86 

88 

90 

93 

96 

99 

102 

1  06 

no 

46 

48 

87 

87 

88 

88 

89 

90 

92 

94 

96 

98 

IOI 

104 

106 

IIO 

114 

118 

48  , 

50 

96 

96 

96 

97 

98 

99 

101 

102 

104 

107 

109 

112 

"5 

119 

123 

127 

50  ' 

52 

105 

105 

106 

106 

107 

109 

IIO 

112 

114 

116 

119 

122 

125 

128 

132 

136 

52 

54 

"5 

"5 

116 

116 

117 

119 

120 

122 

124 

126 

129 

I32 

135 

138 

142 

146 

54 

56 

126 

126 

127 

127 

128 

129 

I3I 

133 

135 

137 

140 

H3 

146 

149 

'53 

157 

56 

58 

138 

138 

138 

139 

140 

141 

H3 

144 

146 

149 

151 

154 

157 

161 

165 

169 

58 

60 
62 

n: 

55 

iS 

l65 

III 

'54 
167 

'I5 
169 

157 
171 

159 

161 
175 

164 
178 

I67 

181 

170 

174 
187 

178 
191 

182 
J95 

60 
62 

64 

179 

179 

180 

181 

182 

184 

I8S 

187 

190 

192 

195 

199 

202 

206 

2IO 

61 

66 

'95 

196 

196 

197 

199 

200 

202 

204 

206 

209 

212 

215 

218 

222 

226 

66 

68 

213 

213 

214 

214 

215 

217 

218 

220 

222 

224 

227 

230 

233 

236 

240 

244 

68 

70 

233 

233 

233 

234 

235 

236 

238 

239 

241 

244 

246 

249 

253 

256 

260 

264 

70 

72 

255 

256 

257 

25« 

260 

26l 

263 

266 

268 

271 

275 

278 

282 

286 

72 

74 

280 

280 

280 

281 

282 

283 

285 

286 

288 

291 

293 

296 

299 

303 

3°7 

3" 

71 

76 

308 

308 

3°9 

3°9 

310 

3" 

315 

317 

319 

322 

324 

328 

33  i 

335 

339 

76 

78 

34i 

34i 

342 

343 

344 

346 

347 

349 

352 

354 

357 

361 

364 

368 

372 

78 

1  80 

380 

380 

381 

382 

383 

384 

385 

387 

389 

39i 

394 

397 

400 

404 

408 

411 

NO 

82 

428 

428 

429 

429 

43° 

43  i 

433 

435 

437 

439 

442 

445 

448 

45i 

455 

459 

82 

84 

490 

491 

491 

491 

492 

494 

495 

497 

499 

5O1 

5°4 

507 

51° 

5*3 

5i7 

52i 

84 

86 

578 

578 

579 

579 

580 

581 

583 

585 

587 

589 

592 

595 

598 

601 

605 

609 

86 

88 

729 

729 

729 

73° 

731 

732 

733 

735 

737 

739 

742 

745 

748 

752 

756 

759 

88 

Entering  Part  I  with  the  Lat.  and  Alt.,  and  Part  II  with  the  Dec.  and  Diflf.  of  the 

Lat.  and  Alt.,  take  out  the  corresponding  numbers.    Then,  with  the  Sum  of  these 

two  numbers,  the  corresponding  Azimuth  is  found  in  Part  HI. 

TABLE  XLI. 


145 


Altitude-Azifiimtlis:  Part  I. 

Alt. 

Latitude. 

Alt. 

.         j 

3O    32 

34° 

36° 

38° 

40° 

42° 

44° 

46° 

48° 

50° 

52° 

54° 

56° 

58° 

6O° 

0 

O 

31  I    36 

41 

46 

52 

58 

64 

7I 

79 

87 

06 

I05 

"5 

126 

138 

150 

0 

O 

2 

31 

36 

41 

46 

52 

58 

64 

71 

79 

87 

96 

105 

"5 

126 

138 

2 

4 

6 

32 

32 

3^ 

37 

42 

46 
47 

52 
53 

58 
59 

a 

72 
73 

80 

80 

88 
88 

96 

1  06 
106 

116 
116 

127 

127 

139 
139 

152 

4 

6 

8 

33 

38 

43 

48 

54 

60 

67 

74 

81 

89 

98 

107 

117 

128 

140 

153 

8 

1O 

34 

39 

44 

49 

55 

61 

68 

75 

82 

90 

99 

109 

119 

129 

141 

i54 

1O 

12 

36 

46 

57 

63 

69 

76 

,    84 

92 

101 

no 

120 

'31 

143 

155 

12 

14 

38 

42 

47 

S2 

58 

64 

78 

86 

94 

102 

112 

122 

133 

144 

157 

14 

16 

4° 

44 

49 

54 

60 

66 

73 

80 

88 

96 

IO4 

114 

124 

135 

146 

159 

16 

18 

42 

47 

52 

57 

63 

69 

75 

82 

90 

98 

107 

116 

126 

137 

149 

161 

18 

20 

45 

49 

54 

59 

65 

7i 

78' 

85 

93 

101 

109 

119 

129 

140 

151 

164 

20 

22 

48 

52 

57 

62 

68 

74 

81 

88 

96 

104 

112 

122 

I32 

H3 

154 

167 

22 

24 

51 

55 

60 

66 

71 

77 

84 

91 

99 

107 

"5 

125 

'35 

146 

158 

170 

24 

26 

54 

59 

64 

69 

75 

81 

88 

95 

102 

HO 

119 

128 

138 

149 

161 

'74 

26 

28 

63 

68 

73 

79 

85 

92 

99 

1  06 

II4 

123 

I32 

142 

153 

165 

178 

28 

30 

62 

67 

72 

77 

83 

89 

96 

103 

no 

118 

127 

I36 

147 

157 

169 

182 

30 

32 

67 

72 

77 

82 

88 

94 

100 

107 

115 

123 

132 

141 

151 

162 

174 

186 

3  ' 

34 

72 

76 

81 

87 

92 

98 

I05 

112 

120 

128 

137 

146 

156 

167 

179 

191 

34 

36 

77 

82 

87 

92 

98 

104 

HO 

117 

J25 

133 

142 

151 

161 

172 

184 

196 

36 

38 

83 

88 

93 

98 

104 

no 

116 

I23 

139 

148 

157 

167 

178 

190 

202 

38 

40 

89 

94 

99 

104 

IIO 

116 

122 

I29 

:37 

H5 

154 

I63 

173 

184 

196 

208 

40 

42 

96 

100 

I05 

no 

116 

122 

129 

I36 

144 

161 

170    180 

191 

202 

215 

42 

44 

103 

107 

112 

117 

123 

I29 

I36 

J43 

'59 

1  68 

177    187 

198 

209 

222 

44 

46 

no 

"5 

120 

J25 

J37 

H3 

158 

1  66 

176 

1841  194 

205 

217 

230 

46 

48 

118 

123 

128 

139 

152 

'59 

166 

174 

184 

192 

202 

213 

225 

238 

48 

50 

127 

132 

J37 

I42 

148 

154 

161 

168 

i;5 

183 

192 

201 

211 

222 

234 

246 

50 

52 

136 

141 

146 

I5I 

157 

163 

170 

177 

184 

192 

201 

211 

221 

231 

243 

255 

52 

54 

146 

151 

156 

161 

167 

173 

180 

187 

194 

202 

211 

221 

23I 

24I 

253 

266 

54 

56 

58 

'57 
169 

162 

174 

167 

179 

172 
184 

178 
190 

184 
196 

191 

202 

198 

209 

205 
217 

2I3 

225 

222 
233 

23I 
243 

242 
253 

264 

264 
275 

277 

288 

56 

58     i 

60 

182 

186 

191 

196 

202 

208 

215 

222 

230 

238 

246 

256 

266 

277 

288 

301 

6O 

62 

J95 

200 

205 

210 

216 

222 

229 

236 

243 

251 

260 

269 

280 

291 

302 

315 

62 

64 

2IO 

215 

220 

225 

23I 

237 

244 

251 

258 

266 

275 

284 

295 

306 

317 

330 

64 

66 

226 

231 

236 

24I 

247 

253 

260 

267 

274 

283 

291 

3OI 

323 

334 

66 

68 

244 

249 

254 

259 

265 

271 

277 

285 

292 

300 

309 

319 

329 

341 

68 

70 

264 

269 

274 

279 

285 

291 

297 

305 

312 

320 

329 

338 

349 

7O 

72 

286 

291 

296 

301 

3°7 

313 

327 

334 

342 

351 

360 

72 

74 

3" 

321 

326 

332 

338 

344 

352 

359 

367 

376 

74 

76 

339 

344 

349 

354 

300 

366 

372 

380 

387 

395 

76 

78 

372 

377 

382 

387 

393 

399 

405 

413 

420 

78 

80 

411 

416 

421 

426 

432 

438 

445 

452 

80 

82 

459 

464 

469 

474 

480 

486 

493 

82 

84 

52i 

526 

531 

536 

542 

548 

84 

86 

609 

614 

619 

624 

630 

86 

88 

759 

764 

769 

774 

88 

Mark  the  Azimuth  N  or  S  according  to  the  Latitude,  and  E  or  W  according  as 

the  object  observed  is  East  or  West  of  the  Meridian. 

When  the  Lat.  exceeds  60°  use  Alt.  for  Lat.  and  Lat.  for  Alt. 

1 

146 


TABLE  XLI. 


Altitude-  Azimuths:   Part  II. 

Diff. 

Declination  of  same  name  as  the  Latitude. 

Diff. 

Alt. 

hM  Jl 

Alt. 

and 

i 

and 

L:it. 

0°      2        i       6 

8° 

10° 

12°  143 

16° 

18° 

20° 

22°  24° 

26    28 

30 

Lat. 

o 

O 

849     842    834  !  825 

817 

808 

799    789 

779 

769 

759 

748    736 

724    712 

699 

0 

2 

849     842    834    825^- 

817 

808 

799     7*9 

779 

769 

758 

748:  736 

724     712 

699 

2 

4 

849      84I     833     825  - 

816 

807 

798     789 

779 

768 

758 

747    735 

698 

4 

6 

848    840  i  832   824 

815 

806 

797:  788 

778 

767 

757 

746    734 

722    710 

697 

6 

8 

847     839    831  :  823 

814 

805 

796;  786 

776 

766 

755 

744    732 

720    708 

695 

8 

10 

846  l  838  1  830  :  822 

813 

804 

796  !  785 

775 

764 

753 

742    73° 

718    705 

692 

1O 

12 

845     837    829!  820  j 

8n 

802 

793     783 

773 

762 

75i 

740    728 

715  !  702 

689 

12 

14 

843  ;  835  1  827    818  i 

809 

800 

791     781 

770    760 

748 

737    725 

712!  699 

685 

14 

16 

841      833!  824    816! 

807 

798 

788;  778 

767 

757 

745 

734    72i 

708   695 

68? 

16 

18 

838  ;  830;  822;  813! 

804 

795 

785    775 

764 

753 

742 

73o!  7i7 

704   691 

677 

18 

20 

836     828!  819    810! 

801 

792 

782  :  771 

t  760    749 

738 

726    713 

699    686; 

672 

2O 

22 

833     825    816:807! 

798 

788 

778    767 

756    745 

733 

721     708 

694    680 

665 

22 

24 

830     821     812  !  803! 

794 

784 

774     763 

752    740 

728 

715 

702 

688    673  i 

658 

24 

26 

826     818    808  ;  7991 

790 

780 

769     758 

747    735 

722 

709 

695 

68  1 

666: 

650 

26 

28 

822     814    804  :  795  j 

785 

775 

764'  753 

7 

4i 

729 

716 

702    688 

673 

658 

641 

28 

30 

818  ;  809    Soo    790 

780 

769 

758     747 

735 

'723 

709 

695    680 

665    649  ; 

631 

30 

32 

814  i  804  |  795     785  i 

775 

764 

752     741 

729 

716 

702 

687    672 

656 

639 

620 

32 

34 

809  ,  799    790    779  ; 

769 

758 

746i  734 

721 

708 

693 

678    662 

645 

627 

608 

34 

36 

803  j  7941  784    773! 

762 

739  j  726 

i  ^ 

699 

684 

668 

651 

634 

595 

36 

38 

797 

788    778    767) 

755 

744 

731     718 

704 

689 

674 

657 

640 

621 

60  1  ! 

579 

38 

40 

791 

781 

771    760 

748 

736 

723 

709 

695 

679 

663 

646 

627 

607 

585: 

562 

40 

42 
44 

785 
778 

774 
767 

764    753 
756    744 

740 

73  i 

727 

7H 
704 

684 
673 

668 
656 

65I 

638 

a 

613 
597 

574 

568! 

548! 

542 
520 

42 
44 

46 

770 

759 

747 

735 

722 

708 

693 

677 

660 

642 

623 

602 

579 

554 

526 

494 

46 

48 

762 

75° 

738 

725 

711 

697 

68  1 

665 

646 

628 

607 

584 

558 

532 

500 

464 

48  ! 

50 

753 

741 

729 

715 

700 

685 

669 

651 

632 

611 

,88 

563 

535 

5°5 

469 

427 

5O 

'52 

744 

732 

718 

704 

688 

672 

655 

636 

615 

593 

568 

540 

5°9 

474 

432  i 

52 

54 

734 

721 

707 

691 

675 

658 

639 

619 

597 

572 

545 

478 

437 

3861 

321 

54 

56 

723 

709 

694 

678 

66  1 

642 

622 

600 

576 

548 

518 

482 

441 

390 

326! 

236 

56 

58 

711 

696 

681 

664 

645 

625 

603 

579 

552 

521 

486 

445 

395 

33° 

240 

87 

58  : 

60 

698  j  683 

666 

648; 

628 

606 

582 

555 

525 

489 

449 

399 

334 

244 

92 

60 

62 

685     668 

650 

631 

609 

585 

558 

528 

493 

452 

402 

338 

248 

96 

62 

64 

670     652 

633 

611 

587 

561 

496 

455 

405 

341 

252 

99 

64 

66 

654     635 

613 

589 

563 

533 

499!  458 

A 

,08 

344 

255 

103 

66  ! 

68 

636 

615 

565 

535 

460    411 

347 

258 

106 

68  i 

70 

616 

593 

567 

537 

503 

462 

413 

350 

260 

109 

70  i 

72 

594  !  568 

539 

505 

464 

415 

352 

263 

in 

72 

74 

569     540 

506 

466 

417 

354 

265     113 

74 

76 

541  !  507 

467 

418 

355 

267 

76 

78 

508     468 

419 

357 

268 

117 

78 

80 

469 

420 

358 

269, 

118 

80  > 

82 

421 

358 

270 

119 

82 

84 

359 

271 

1  20 

84  j 

86 

2.71 

121 

86  | 

88 

121 

88  | 

Altitude-  Azimuths  :  Part  III. 

1        « 

a     ^ 

a 

rf 

a 

ti 

a 

M 

a 

si 

a 

2          Kj 

a     M. 

*        i 

10 

<l 

a 

<t 

CO 

<1 

CO 

<! 

Si 

<J 

CQ 

<J 

CQ      <! 

•Ji      <S 

n 

o 

o 

o 

o 

o 

o 

o 

240 

20 

413 

30 

534 

40 

626 

50 

699 

60 

759    70 

8O8    80 

85O    90 

261 

21 

427 

544 

631 

51 

7O5 

61 

764    71 

813    81 

851    91 

19     12 

2N1       22 

440 

32 

554 

42 

612 

52 

712 

62 

769    72 

817    82 

857    92 

54    n 

300     23 

453 

33 

564 

43 

650 

53 

718 

63 

774    73 

821    83 

861    93 

86 

14 

318 

24 

466 

34 

574 

44 

657 

54 

724 

64 

779    74 

826    84 

864    94 

116 

!5 

335 

25 

478 

35 

583 

'  45 

664 

55 

730 

65 

784    75 

830    85 

868    95 

144 

16 

352 

26 

490 

36 

592 

46 

672 

5" 

736 

66 

789    76 

834    86 

871    96 

17O 

17 

368 

27 

501 

37 

601 

47 

679 

57 

742 

67 

794    77 

838    87 

874    97 

194 

18 

3S4 

28 

512 

38 

6O9 

48 

686 

58 

748 

68 

799    78 

842    88 

878    98 

218 

19 

399 

29 

523 

39 

618 

49 

692 

59 

753 

69 

804    79 

846    89 

881    99 

TABLE  XLI. 


147 


Attitude-Azimuths:  Part  II. 

iDifT 

Declination  of  contrary  name  to  the  Latitude. 

Diff.l 

Alt. 

„  .3 

Alt. 

ori  Jl 

ana 
Lat. 

O° 

2° 

4° 

6° 

8° 

10° 

12° 

14° 

16° 

18° 

2O3 

22° 

24° 

26° 

28° 

30° 

ana 
Lat. 

o 

0 

849 

857 

864 

871 

878 

884 

890 

896 

902 

908 

913 

918 

923 

928 

933 

937 

o 

O 

2 

849 

857 

864 

871 

878 

884 

890 

896 

902 

908 

913 

918 

923 

928 

933 

937 

2 

4 

849 

856 

864 

871 

877 

884 

890 

896 

902 

907 

913 

918 

923 

928 

933 

937 

4 

6 

848 

856 

863 

870 

877 

883 

889 

895 

901 

907 

912 

918 

923 

928 

933 

937 

6 

8 

847 

855 

862 

869 

876 

882 

888 

895 

901 

906 

912 

917 

922 

927 

932 

936 

8 

10 

846 

854 

86  1 

868 

875 

88  1 

887 

894 

900 

905 

911 

916 

921 

926    931 

935 

10 

12 

845 

852 

859 

867 

874 

880 

886 

893 

899 

904 

910 

9J5 

920 

925    93° 

934 

12 

14 

843 

850 

858 

865 

872 

879 

885 

891 

897 

903 

909 

914 

919 

924    929 

933 

14 

16 

841     848 

856 

863 

870 

877 

883 

890 

896 

901 

907 

912 

917 

922    927 

932 

16 

18 

838 

846 

854 

861 

86? 

I 

875 

88  1 

888 

894 

900 

9°5 

911 

916 

921 

926 

930 

18 

2O 

836 

844 

851 

859 

866 

873 

879 

886 

892 

898 

903 

909 

914 

919 

924 

929 

20 

22 

833 

841 

849 

856 

863 

870 

877 

883 

889 

895 

901 

907 

912 

917 

922 

927 

22 

24 

830 

838 

8 

46 

853 

860 

867 

874 

88  1 

887 

893 

899 

904 

910 

915 

920 

925 

24 

26 

826 

834 

8 

42 

850 

857 

864 

871 

878 

884 

896 

901 

907 

913 

917 

922 

26 

28 

822 

831 

8 

39 

847 

854 

86  1 

868 

875 

88  1 

887 

893 

899 

905 

910 

915 

920 

28 

30 

818 

827 

835 

843 

851 

858 

865 

872 

878 

884 

890 

896 

902 

907 

912 

917 

30 

32 

814 

823 

831 

839 

847 

854 

861 

868 

875 

88  1 

887 

893 

898 

904 

909 

914 

32 

34 

809 

818 

826 

834 

842 

850 

857 

864 

871 

877 

884 

890 

895 

901 

906 

911 

34 

36 

804 

813 

821 

830 

83' 

J 

846 

853 

860 

867 

874 

880 

886 

892 

897 

903 

908 

36 

38 

798 

807 

816 

825 

833 

841 

849 

856 

863 

870 

876. 

882 

888 

894 

899 

904 

38 

40 

792     80  1 

810 

819 

828 

836 

844 

851 

858 

865 

872 

878 

884 

890 

8( 

^ 

901 

40 

42 

785      795 

804 

813 

821 

830 

839 

846 

853 

860 

867 

874 

880 

886 

891 

897 

42 

44 

778 

788 

798 

807 

816 

825 

833 

841 

848 

855 

862 

869 

875 

881 

887 

892 

44 

46 

770 

781 

791 

801 

810 

819 

827 

835 

843 

850 

8,57 

864 

870 

877 

882 

888 

46 

48 

762 

773 

784 

794 

803 

812 

821 

829 

837 

845 

852 

859 

865 

872 

878 

883 

48 

50 

753 

765 

776 

786 

79( 

) 

805 

814 

823 

831 

839 

846 

853 

860 

866 

873 

878 

50 

52 

744 

756 

767 

778 

78 

J 

797 

807 

816 

824 

832 

840 

847 

854 

86  1 

867 

873 

52 

54 

734 

747 

758 

769 

780 

790 

800 

809 

818 

826 

834 

841 

848 

855 

862 

868 

54 

56 

723 

736 

749 

760 

771 

782 

792 

801 

810 

819 

827 

835 

842 

849 

856 

862 

56 

58 

711 

725 

738 

7 

5i 

762 

773 

783 

793 

802 

811 

820 

828 

835 

842 

850 

856 

58 

60 

698 

713 

727 

740 

752 

764 

774 

784 

794 

803 

812 

820 

828 

836 

843 

60 

62 

685 

700 

715 

729 

742 

754 

765 

775 

785 

795 

804 

812 

821 

828 

62 

64 

670 

687 

702 

717 

730 

743 

755 

766 

776 

7.86 

796 

804 

813 

64 

66 

654 

672 

688 

704 

718 

73  * 

744 

755 

766 

777 

787 

796 

66 

68 

636 

656 

673 

690 

705 

719 

732 

744 

756 

767 

777 

68 

70 

616 

638 

657 

674 

691 

7°5 

719 

732 

745 

756 

70 

72 

594 

618 

639 

658 

67. 

5 

691 

706 

720 

733 

72 

74 

569 

595 

619 

639 

65* 

* 

676 

692 

706 

74 

76 

54i 

57o 

596 

619 

640 

659 

676 

76 

78 

508 

542 

57i 

596 

620 

640 

78 

80 

469 

509 

543 

57i 

599 

8O 

82 

421 

470 

509 

543 

82 

84 

359 

422 

470 

84 

86 

271 

359 

86 

88 

121 

88 

Altitude-Azimuths  :  Part  III. 

d 

. 

a 

g 

s 

a 

d 

d 

g 

rj 

N 

3 

Nl 

rj 

N 

3 

t-J 

3 

N 

f-t 

N 

9 

o 

5 

fc-3 

02 

<j 

02 

<J 

02 

02 

0 

02 

m 

<J 

02 

<J 

02 

< 

884 

0 

100 

913 

o 

no 

938 

o 
120 

957 

0 

130 

973 

o 
140 

985 

0 

993 

o 
1  60 

998 

0 

170 

887 

101 

916 

III 

94O 

121 

959 

974 

141 

986 

jCj 

994 

161 

999 

171 

890 

I  O2 

919 

112 

942 

122 

961 

132 

976 

142 

987 

I"?2 

995 

162 

999 

/  * 
172 

894 
897 

I03 
IO4 

921 
924 

"3 

114 

944 
946 

I23 
124 

962 
964 

133 
134 

977 

978 

143 
144 

988 
989 

\H 

995 
996 

163 
164 

999 
999 

?  /  . 

173 

174 

899 

105 

926 

115 

948 

125 

966 

*35 

979 

J45 

99O 

'55 

996 

165 

1OOO 

I7"5 

9O2 

106 

928 

116 

95O 

126 

967 

136 

981 

•35 

146 

990 

997 

166 

1OOO 

176 

9O5 
908 

107 
1  08 

931 

933 

117 
118 

952 
954 

127 
128 

969 
970 

137 

138 

982 
983 

H7 
148 

991 
992 

158 

997 

998 

167 
1  68 

1OOO 
1OOO 

177 
178 

911 

109 

935 

119 

955 

129 

972 

139 

984 

149 

993 

998 

169 

1000 

1  80 

148 


TABLE  XLII. 


Altitude-Azimuths  :   Direct  and  Limiting  Values. 

Lat. 

Altitude. 

Lat. 

O° 

10° 

20° 

30° 

40° 

50° 

603 

70° 

80° 

9OD 

0 

0 
10 
20 
30 
40 
50 
6O 
70 
80 

0 

0 
10 
20 
30 
40 
5O 
60 
70 
§0 

o 

0 
10 
20 
30 
40 
50 
60 
7O 
80 

o 

0 
10 

20 
30 
40 
50 
6O 
70 
80 

0 

O 
10 
20 
30 
40 
50 
60 
70 
80 

Pol.  Dist.  60°,  or  Declination  30°  of  same  name  as  the  Latitude. 

o   ! 
O 
10 
20 
30 
40 
50 
60 
7O 
80 

o 

O 
10 
20 
3O 
40 
50 
6O 
70 
8O 

0 

O 
10 
20 
3O 
40 
50 
6O   j 
70 
80 

o 

0  i 

10 
2O 

3O   i 
40 
50 
6O 
70 
80   | 

o 

0 

10 

20 
30 
40 
5O 
6O 
70  I 
8O  ! 

,  o 

oo.o 
59-5 
57-9 
54-7 
49-3 
38.9 
o.o 

o 

59-5 
61.0 

61.5 

61.0 
59-o 
54.6 
44-8 
o.o 

0 

57-9 
61.5 

64-3 
66.2 
67.1 
66.8 

64-3 
56.2 
o/o 

0 

54.7 

61.0 
66.2 
70.5 
74-4 
77-9 
fti.'i 
84.1 
87.1 

0 

49-3 
59-0 
67.1 

74-4 
81.5 
89.1 
98.5 
"3-4 
180.0 

o 

38.9 

54-6 
66.8 

77-9 
89.1 

102.  1 
I2O.6 
ISO.O 

o 
O.O 
44-8 
64.3 

81.1 
98.5 

120.6 

180.0 

o 
O.O 
56.2 
84.1 

II3-4 
I  So.O 

o 
O.O 

87.1 

180.0 

Indet. 

Pol.  Dist.  70°,  or  Decimation  20°  of  same  name. 

0 

70.0 

69.7 

68.6 
66.7 
63.5 
57-9 
46.8 
o.o 

69.7 
71.2 
72.2 
72.6 
72.2 
70.7 
67.1 

57-9 
o.o 

68°.6 

72.2 
75-2 
77-9 
80.2 
82.4 
84.4 
86.3 
88.2 

66°7 
72.6 

77-9 
82.9 
88.2 
94.2 

102.  1 
II5.6 

ISO.O 

63.5 

72.2 
80.2 

88.2 
96.9 
107.8 
124.1 
1  80.0 

0 

57-9 
70.7 
82.4 
94.2 
107.8 
126.3 
ISO.O 

46°.8 
67.1 
84.4 

1  02.  1 
I24.I 

1  80.0 

0 

0.0 

57.9 

86.3 

115.6 
180.0 

o 

o.o 
88.2 
180.0 

ludet. 

Pol.  Dist.  80°,  or  Declination  10°  of  same  name. 

o 

80.0 

79.8 

79.3 
78.4 
76.9 

74.3 
69.7 

59-5 
o.o 

o 

79-8 
81.5 
82.9 
84.1 

85-3 
86.3 

87-3 
88.2 
89.1 

0 

82.9 

86.3» 
89.8 
93-7 
98.4 
105.1 
117.4 
iSo.o 

78°-4 
84.1 
89.8 

95-8 
102.9 

112.  1 

126.8 
ISO.O 

76.9 
85.3 
93-7 
102.9 
114.1 

130-3 
180.0 

o 

74-3 
86.3 
98.4 

II2.I 

130.3 
1  80.0 

o 
697 
87.3 
I05.I 
126.8 
ISO.O 

o 
1174 

180.0 

o 
O.O 
89.I 
ISO.O 

Indet. 

Pol.  Dist.  90°,  or  Declination  0°. 

o 

90.0 
90.0 
90.0 
90.0 
90.0 
90.0 
90.0 
90.0 
90.0 

•o 
90.0 
90.8 

93-7 
95-8 
98.5 

IO2.I 
107.8 
II9.O 
1  80.0 

0 

90.0 

93-7 
97.6 

I02.I 
107.8 

II5-7 
I29.I 
ISO.O 

o 

90.0 
95-8 

102.  1 
I09.5 
II9.0 

133-5 
ISO.O 

o 

90.0 

98.5 

107.8 

119.0 

134.8 

180.0 

o 

90.0 
1  02.  1 

115.7 
133.5 

1  80.0 

0 

9O.Q 
107.8 
I29.I 
ISO.O 

0 

90.0 
II9.0 
ISO.O 

0 

90.0 
ISO.O 

Indet. 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name. 

1  00.0 
IOO.2 
100.7 

101.6 

103.1 

105.7 
110.3 
120.5 
180.0 

0 

100.2 
1  02.  1 
104.6 
107.8 
1  12.2 
II9.0 
131.1 
ISO.O 

0 

100.7 
104.6 
IO9.2 

IIS-I 
I23.I 
136.2 
1  80.0 

0- 

101.6 
107.8 

115-1 
124.4 

138-3 
1  80.0 

0 

103.1 

1  12.2 
I23.I 

138.3 
1  80.0 

0 

105.7 

119.0 
136.2 
1  80.0 

o 
110.3 

131.1 

180.0 
i 

o 
I2O.5 
1  80.0 

i8o°.o 

TABLE  XLIII. 


149 


Position-Angles  for  Direct  and  Limiting-  Altitude-  Azimuths. 

!  Lat. 

Altitude. 

Lat. 

O°          1O° 

20° 

303        40° 

5O° 

60° 

70° 

80° 

90° 

o 

L  o 

10 

20 

3O 
4O 
50 
6O 
7O 
80 

o 

0 

10 

20 
30 
4O 
50 
6O 
70 
80 

o 

O 
10 

20 
3O 
40 
50 
60 
70 
8O 

o 

0 
10 

20 

30 
40 
5O 
60 
70 
8O 

o 

0 
10 
20 
3O 
40 
50 
6O 
70 
80 

Pol.  Dist.  60°,  or  Declination  30°  of  same  name  as  the  Latitude. 

0 

0 
10 
20 
30 
40 
5O 
60 
70 
80 

o 

0 
10 

20 
30 
40 

5O   ! 
60   ! 
70 

80 

o 

O 
10 

20 
30 
40 
50 
60 
70 
80 

o 
0 
10 
20 
30 
40 
50 
60 
70 
80 

o 

0 
10 
20 
30 
40 
50 
6O 
70 
80 

0 

90.0 
78.4 
66.7 

54-7 
42.1 
27.8 
o.o 

o 

95-8 
84.0 
72.5 
61.0 
49-3 
37-2 
24.0 

0.0 

0 

102.2 
88.2 

77-3 
66.2 

54-5 
43-o 
3i-3 

19.2 
o.o 

o 

109.5 

96.0 

83.1 
70.5 
58.4 
46.5 

34-8 
23.1 
1  1.6 

0 

II9.0 
IO2.9 

88.2 

74-3 
61.0 

47-9 
34-8 

21.2 
0.0 

o 

133-5 

112.  1 
94.1 

77-9 
62.2 

46-5 
29.8 
o.o 

0 

180.0 

126.8 

102.  1 

81.1 
61.0 

39-7 
o.o 

180.0 
115.6 
84.1 

54-3 
o.o 

i8o°.o 
87.0 
o.o 

Indet. 

Pol.  Dist.  703,  or  Declination  20°  of  same  name. 

o 

.  90.0 

79-4 
68.7 
57.8 
46.8 

35-4 
22.9 
o.o 

o 

93-7 
82.7 
72.2 
61.5 

5°-9 
40.2 
29.3 
18.0 
o.o 

o 
97.6 

86.1 
75-2 
64-3 
534 
42.7 
32.0 
21.3 
10.6 

o 
102.  1 

88.8 

77-9 
66.1 

54-6 
43-° 
3i-3 
19.2 
o.o 

io7°.8 

93-9 
80.2 
67.1 
54.0 
40.6 
26.1 
o.o 

0 

U5-7 

98.5 
82.4 

66.8 
5°-9 
334 

o.o 

o 
I29.I 
105.2 
84.4 

64-3 

42.4 
o.o 

i8o°.o 
117.4 
86.3 
56-2 

0.0 

o 
l8o.O 
88.3 
O.O 

• 

luclet. 

. 

Pol.  Dist.  80°,  or  Declination  10°  of  same  name. 

o 

90.0 

79-9 
69.7 

59-5 
49-3 

284 
17.4 
o.o 

0 

91.7 
8l.5 

T3 
OI.O 

50-5 

40.6 

30-5 
20.3 

IO.2 

0 

93-7 
82.9 
72.2 
61.6 

50-9 
40.2 
29.3 

o.o 

95-8 
84.1 
72.6 
61.0 
48.1 
37-2 
24.0 
o.o 

08.5 

85.2 
72.2 

59-o 

45-3 
29.8 
o.o 

o 
IO2.I 
86.3 
70.7 

54.6 

3^-3 
o.o 

io7°.8 

87-3 
67.1 

44-8 
o.o 

0 

II9.0 

88.3 

57-9 
o.o 

1  80.0 
89.1 

0.0 

Indet. 

Pol.  Dist.  90°,  or  Declination  0°. 

0 

90.0 

80.0 

70.0 
60.0 
50.0 
40.0 
30.0 

2O.O 
10.0 

o 
9O.O 
80.0 
69-3 

59-5 
49-3 
38.0 

28.4 
17.4 
o.o 

o 
9O.O 

79-3 
68.7 
57.8 
46.8 
35-4 

22.8 
0.0 

o 
9O.O 
78.4 
66.7 

54-7 
42.1 
27.8 
o.o 

0 

90.0 
76.9 
63-5 

49-3 
32-9 
o.o 

0 

90.0 

74-3 
57-9 
38-9 
o.o 

o 
9O.O 
69.7 
46.8 
O.O 

o 
9O.O 

59-5 
o.o 

0 

90.0 
0.0 

Indet. 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name. 

0 

90.0 

79-9 
69.7 
59-5 
49-3 
38-9 
28.4 
17.4 
o.o 

o 
88.3 

77-9 
67.5 
56-9 
46.1 

34-8 
22.5 
o.o 

o 

86.3 

75-4 
64-3 
52.8 
40.6 
26.9 
o.o 

0 

84.2 
72-3 
59-8 
46.5 
31.2 
O.O 

8i°.5 
67.8 
53-o 
35-8 
o.o 

o 

77-9 
61.0 
41.4 
o.o 

0 

72.2 
48.9 
O.O 

,  ° 

61.0 
o.o 

o 
O.O 

150 


TABLE  XLII. 


Altitude-Azimuths:  Direct  and  Limiting  Values. 

T  ^*- 

Altitude. 

Lat 

,  Jbat. 

0° 

10° 

20° 

3O° 

40° 

50° 

6O° 

7O°    |    8O° 

90° 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name  to  Latitude. 

o 

o 

o 

o 

o 

o 

0 

0 

0 

0 

0 

0 

1  00.0 

IOO.2 

IOO.7 

101.6 

I03.I 

1057 

110.3 

120.5 

180.0 

o 

10 

100.2 

1  02.  1 

104.6 

107.8 

1  12.2 

II9.0 

131.1 

180.0 

10 

20 

100-7 

104.6 

109.2 

115.1 

I23.I 

136.2 

1  80.0 

20 

30 

101.6 

107.8 

II5.I 

124.4 

138.3 

1  80.0 

3O 

i  40 

103.1 

II2.2 

I23.I 

138.3 

1  80.0 

40 

!   50 
60 
70 

105.7 
110.3 
120.5 

II9.0 
1  80.0 

136.2 
1  80.0 

180.0 

5O  • 
60 
7O 

80 

180.0 

80 

Pol.  Dist.  110°,  or  Declination  20°  of  contrary  name. 

o 

o 

o 

o 

o 

o 

0 

1  1  0.0 

IIO-3 

III.4 

II3-3 

116.5 

I22.I 

133-2 

180.0 

0 

10 

110.3 

1  12.6 

II5-7 

I2O.2 

126.9 

138.6 

1  80.0 

10 

20 

111.4 

115-7 

I2I.3 

I29.I 

I4I-3 

180.0 

20 

30 

"3-3 

120.2 

I29.I 

I42.I 

1  80.0 

30 

40 

116.5 

126.9 

141.3 

1  80.0 

4O 

50 

1  22.  1 

138.6 

1  80.0 

50 

60 

133-2 

1  80.0 

60 

70 

180.0 

7O 

Pol.  Dist.  120°,  or  Declination  30°  of  contrary  name. 

0 

o 

o 

o 

0 

o 

0 

o 

o 

O 

120.0 

120.5 

I22.I 

125.3 

130-7 

I4I.I 

180.0 

o 

10 

120.5 

123.1 

127.2 

133-5 

144.2 

1  80.0 

1O     1 

20 

122.  1 

127.2 

134.3 

145-5 

180.0 

2O     | 

30 

125-3 

133-5 

H5.5 

180.0 

3O  ! 

40 

130.7 

144.2 

ISO.O 

4O 

50 

I4I.I 

180.0 

50 

60 

1  80.0 

60 

TABLE  XLIV. 

Error  of  the  Altitude-Azimuth  for  an  Error  of  6',  or  O°.l,in  the  Altitude.  I 

Pos. 

Altitude. 

Pos. 

Ang. 

O° 

10° 

20° 

30° 

40° 

50° 

60° 

70° 

75° 

80° 

82° 

84° 

86° 

88° 

90 

Ang.: 

o 

90 

o' 

o' 

o' 

o' 

0' 

o' 

o' 

o' 

0 

0.0 

0 

0.0 

o 
0.0 

o 
0.0 

0 

0.0 

o 
O.O 

Indet. 

o 

9O 

85 

i 

i 

i 

i 

i 

i 

i 

2 

0 

I 

I 

I 

I 

3 

00 

95 

8O 

j 

i 

i 

j 

2 

2 

3 

I 

I 

I 

2 

3 

S 

oo 

100 

75 

2 

2 

2 

2 

2 

3 

3 

5 

I 

2 

2 

3 

4 

8 

00 

1O5 

70 

2 

2 

2 

3 

3 

3 

4 

6 

I 

2 

3 

3 

5 

I.O 

00 

no 

65 

3 

.   3 

3 

4 

4 

6 

9 

0.2 

o-3 

°-3 

0.5 

°-7 

1.3 

00 

115 

60 

4 

4 

4 

4 

5 

7 

10 

2 

3 

4 

6 

8 

7 

00 

12O 

55 

4 

4 

6 

8 

12 

3 

4 

5 

7 

I.O 

2.0 

00 

125 

5O 

c 

c 

5 

7 

8 

10 

15 

3 

S 

6 

8 

2 

4 

00 

13O 

45 

6 

6 

6 

7 

8 

9 

12 

18 

4 

6 

7 

I.O 

4 

9 

oo 

135 

40 

7 

7 

8 

8 

9 

12 

14 

21 

°-5 

0.7 

0.9 

i.i 

*-7 

3-4 

00 

140 

35 

9 

9 

9 

10 

ii 

13 

17 

2S 

6 

8 

I.O 

4 

2.1 

4.1 

00 

145 

30 
25 
2O 

10 

'3 
*7 

II 

13 
17 

ii 

if 

12 
15 
19 

17 

22 

20 
26 

21 
26 

33 

30 
38 
48 

8 
i.i 

I.O 

2 

6 

2 

5 

2.0 

7 

2.O 

6 

5 
9 

5-o 

6.2 

7-9 

00 

oo 

00 

15O 
155 
160 

18 

*9 

19 

20 

21 

24 

29 

37 

54 

1.2 

1.8 

2.2 

3-o 

4-4 

8.8 

162 

16 

21 

21 

22 

24 

27 

33 

42 

61 

3 

2.O 

5 

3 

5-o 

164 

14 
12 

11 

24 
29 

26 

28 

33 

31 

37 

37 
44 

48 

70 
83 

I 

3 
7 

9 
3-4 

8 

166 
168 

10 

34 

35 

36 

39 

44 

53 

68 

100 

2.2 

3-3 

17O  i 

8 

43 

45 

49 

S6 

66 

8S 

125 

2-7 

172 

6 
4 

SI 

87 

91 

66 

75 

112 

89 
'33 

114 

174 
176 

2 

172 

i75 

183 

198 

225 

178 

1  ta  4  k   ' 

O 

00 

00 

00 

00 

J.8V 

TABLE  XLIII. 


151 


Position-  Angles  for  Direct  and  Limiting  Altitude-  Azimuths. 

Lat. 

Altitude. 

Lat. 

0° 

10° 

2O° 

30° 

40° 

5O° 

60° 

70° 

80° 

90° 

0 

O 
10 
2O 
3O 
40 
50 
6O 
70 
§0 

o 

O 
10 
20 
3O 
40 
50 
60 
70 

0 

0 
10 
20 
30 
4O 
1   50 
60 

Pol.  Dist.  100°,  or  Declination  10°  of  contrary  name. 

0 

O 
10 
20 
30 
40 
50 
60 
70 
80 

o 

O 

10 

20 
30 
40 
50 
6O 
70 

0 

0 
10 
20 
30 
40 
50 
60 

o 

90.0 
79-9 
69.7 
59-5 
49-3 
38-9 
28.4 
17.4 

0.0 

88°.8 
77-9 
67.5 
56.9 
46.1 

34-8 
22.5 
o.o 

o 
86.3 

75-4 
64-3 
52.8 
40.6 
26.9 
o.o 

o 
84.2 

72-3 
59.8 

46.5 
31-2 
O.O 

at* 

67.8 

53-o 
35-8 
o.o 

0 

77-9 
61.0 
41.4 
o.o 

o 

72.3 
48.9 

0.0 

o 

61.0 
o.o 

o 
O.O 

Pol.  Dist.  110°,  or  Declination  20°  of  contrary  name. 

o 

90.0 

794 
68.7 

Hi 

35-4 
22.9 
o.o 

86.3 
72.8 
64-3 

40.6 
26.9 
o.o 

&.4 

70.8 
58.7 

45-7 
30.6 
o.o 

o 

77-9 
64.9 

5°-9 
34-5 
o.o 

o 
72.2 
56.9 

38.7 
0.0 

o 
64.3 

43-8 
o.o 

0 

5°-9 
o.o 

o 

o.o 

Pol.  Dist.  120°,  or  Declination  30°  of  contrary  name. 

o 

90.0 

78.4 
66.7 

54-7 
42.1 
27.8 

0.0 

o 
84.2 
72.2 
59.8 

46.5 
31.2 
O.O 

77°S 

64.9 

5°-9 
34-5 
o.o 

0 

70.5 

55-6 
37-9 
o.o 

o 

61.0 
41.7 
o.o 

o 

46-5 
o.o 

0 

0.0 

TABLE  XLV. 

Error  of  the  Altitude-Azimuth  for  an  Error  of  12',  or  O°.2,  in  Latitude. 

Hour- 
Angle. 

Latitude. 

Hour- 
Angle. 

0° 

10° 

2O° 

30° 

4O° 

50° 

60° 

65° 

70° 

72° 

74° 

76° 

78° 

80° 

h    m, 
6     O 
5  40 
20 
O 
4  40 
4  20 
0 
3  4O 
20 
O 
2  40 
20 
O 
1  40 
20 
1  12 
4 
0  56 
48 
40 
0  32 
24 
16 
8 
0 

0 

0.0 
O 

o 
I 

I 

O.I 

I 

2 
2 
O.2 

3 
3 
4 
5 
0.6 

I 

9 
i.i 

1.4 
1.9 
2.9 
5-7 

00 

o 
0.0 
0 

o 

0. 

2 
2 
O.2 

3 

0 

0.0 
0 

o 
I 
I 

O.I 

I 

2 
2 
2 

0-3 

3 

0 

0.0 
O 

o 

I 
I 

O.I 

I 

2 
2 
2 

o-3 
3 

0 

0.0 
0 

I 
I 
I 

O.I 
2 
2 
2 
3 

o-3 

4 

0 

0.0 

o 
I 

I 
I 

O.I 
2 
2 
3 

3 

0.4 

4 

0 

0.0 
O 

I 

O.2 

2 

3 
3 
4 
0.5 

o 
0.0 
O 

I 

2 
O.2 
3 

3 

4 
5 
0.6 

8 

I.O 

3 

1-5 

7 

2.0 
2 

7 
3-4 

o 
0.0 

o 

I 

2 
2 

o-3 
3 
4 

0.7 

I.O 

1.8 

2.0 

3 
7 
3-3 

0 

0.0 

I 
I 
2 
2 

o-3 
4 
5 

0.8 

9 
i.i 

1 

2.0 

3-o 
7 

o 
O.O 

I 

I 

2 

3 
o-3 
4 

7 
0.9 

I.O 

i 

2.0 
2.2 
5 

9 
3-4 

o 
0.0 

I 
I 

2 

3 
0.4 

I.O 
2 

•  i 

2-3 

2-5 

9 
3-3 

0 

0.0 
I 
2 

3 
3 
0.5 

I.O 

i.i 

4 
7 

2.1 

6 

3-o 
4 

0 

0.0 

I 

2 

3 
4 
o-5 

1 

I.O 
2 
1.4 

6 

2.0 

5 

3-2 

3-5 

h     m 
6    0 
20 
40 
7    0 
20 
740 
8    0 
20 
40 
9    O 
920 
40 
10    0 
20 
40 
1048 
56 
11    4 
12 
2O 
1128 
36 
44 
52 
12    O 

4 
6 

0.6 

I.O 
2 
1.4 
1.9 
2.9 

5-8 

I 

0.7 

7 
9 

I.O 
2 

1-5 
2.O 

3'0 

I 

0.7 

9 
i.i 

3 
1.6 

2.2 

3-3 

5 
7 
0.8 
9 

I.O 
2 

5 

1.9 

2-5 

7 
9 

I.O 

i 

2 

t 

2.2 

9 
i.i 

1.2 

I 

9 
2-3 

2.8 

152 


TABLE  XLVL 


I 

Error  of  Altitude-Azimuth  for  an  Error  of  6',  or  O.I.  in  Declination. 

Hour- 

Latitude. 

i 

Hour- 

Angle. 

O° 

10° 

20° 

3O° 

4O° 

5O° 

6O° 

65° 

70° 

72° 

74° 

76° 

78° 

80 

Angle. 

j     h    in 

o 

0 

o 

0 

0 

0 

0 

0 

o 

0 

0 

o 

0 

o 

h     in 

6     0 

O.I 

O.I 

O.I 

O.I 

O.I 

0.2 

0.2 

0.2 

0.3 

0.4 

0.4 

0.5 

0.6 

6    0 

5  40 

I 

2 

2 

2 

3 

3 

4 

4 

S 

6 

20 

20 

I 

2 

2 

2 

3 

3 

4 

4 

S 

6 

40 

0 

I 

2 

2 

2 

3 

3 

4 

4 

S 

6 

7    0 

4  40 

I 

2 

2 

3 

3 

3 

4 

4 

5 

6 

20 

i  4  2O 

0. 

0. 

0. 

O. 

O.I 

O.2 

O.2 

O. 

3 

0-3 

0.4 

0.4 

o-.S 

0.5 

0.6 

740 

0 

2 

2 

2 

3 

3 

4 

4 

S 

6 

7 

8    O 

3  40 

2 

2 

2 

> 

4 

-4 

4 

6 

7 

20 

20 

2 

2 

2 

3 

3 

4 

4 

S 

^ 

6 

8 

40 

O 

2 

2 

2 

2 

3 

3 

4 

5 

5 

6 

7 

8 

9    0 

2  4O 

0.2 

0.2 

0.2 

0.2 

6.2 

0.2 

o-3 

0.4 

0-5 

o-S 

0.6 

0.6 

0.7 

0.9 

920! 

20 

2 

2 

2 

2 

2 

3 

3 

4 

6 

6 

7 

8 

I.O 

40  1 

0 

2 

2 

2 

2 

3 

3 

4 

S 

6 

6 

7 

S 

I.O 

2 

1O    0 

1  40 
*>o 

2 

2 

3 

3 

3 

4 

(> 

T 

7 

8 

9 
i  i 

I.O 
2 

i 

4 

20 

1  12 

0-3 

<M 

0-3 

0.4 

0.4 

0-5 

0.6 

0.8 

I.O 

I.O 

1.2 

1-3 

'  1.6 

1.9 

1O48 

4 

4 

4 

4 

4 

<> 

6 

7 

9 

i 

2 

3 

^ 

7 

2.1 

56 

0  56 

4 

4 

4 

S 

6 

8 

I.O 

2 

4 

5 

7 

2.0 

11    4 

48 

5 

6 

6 

7 

I.O 

i 

4 

6 

7 

2.0 

12 

4O 

6 

6 

6 

7 

8 

9 

2 

4 

7 

8 

2.1 

20 

J  0  32 

0.7 

0.7 

0.8 

0.8 

0.9 

i.i 

1.4 

J. 

7 

2.1 

2-3 

1128 

24 

I.O 

1.0 

I.O 

i. 

i 

1.2 

S 

9 

2-3 

2.8 

36  i 

16 

4 

5 

5 

6 

9 

2.2 

2.9 

3-4 

44 

8 

2.9 

2.9 

3-3 

3-7 

4-4 

5-7 

52 

O 

oo 

00 

00 

12    0 

TABLE  XL  VII. 

Limiting  Errors  of  Altitude-Azimuths. 

Least  Hour-Angle  1  . 

Least  Hour-Angle  2il. 

Least  Hour-  Angle  3h. 

Partial  Az.  Error. 

Partial  Az.  Error. 

Partial  Az.  Error. 

Lat. 

Prob. 

Prob. 

Prob. 

Total 

Total 

Total  1 

Alt 
Error 

Lat.       Dec. 
Error     Error 

Az. 
Error. 

Alt.        Lat. 
Error     Error 

Dec. 
Error 

Az. 
Error. 

Alt. 
Error 

Lat. 
Error 

Dec. 
Error 

Az. 
Error. 

±  6'. 

±  12'.      ±  3'. 

±  6'-       ±  12'. 

±3'. 

±6'. 

±12'. 

±3'. 

o 

0 

0 

±0.4 

0                      0 

±0.8       ±0.2 

0 

±0.9 

0                      0 

±0.2       ±0.4 

o 
±0.1 

o 

±0.5 

o 
±0.1 

0 

±0.2 

0 

±0.1 

0 

±0.3 

10 

0.4 

0.8        0.2 

0.9 

0.2           0.4 

O.I 

o-5 

O.I 

0.2 

O.I 

0.3 

20 

0.4 

0.8           0.2 

I.O 

O.2           0.4 

O.I 

O.I 

0.2 

O.I 

30 

0.9      0.3 

I.I 

0.3           0.4 

0.2 

0.6 

O.I 

0.3 

O.I 

0.4 

40 

°-5 

i.o        0.3 

1.2 

0.3           0.5 

0.2 

0.6 

0.2 

O-3 

O.I 

0.4 

50 

0.6 

1.2           0.3 

i-7 

0.4        0.6 

0.2 

0.8 

0.2 

0.3 

0.2 

°-5 

60 

0.8 

1.6        0.4 

1.9 

0.4        0.7 

O.2 

0.8 

°*3 

0.4 

O.2 

0.6 

70 

i.i 

2.3        0.6 

2.6 

0.6         i.o 

O.  "1 

1.2 

0.4 

0.6 

0.2 

0.8 

80 

2.8 

4-5         1-2 

5-5 

1-3            2.O 

0.6 

2-5 

I.O 

1.2 

0-4 



TABLE  XLVIII. 


153 


Time-Alt.  Azimuths  :  Log. 

A. 

Declination. 

Diff. 

A  r* 

for 

A  *T 

JLZ. 

0° 

5 

10° 

15° 

2O° 

25° 

30° 

35° 

40° 

45° 

50° 

IvJ-i 

J*.Z.   , 

i 

i 

o 

o 

+  00 

+  00 

+  00 

+ 

oo 

+  00 

+  00 

+  00 

+ 

00 

JL 

00 

+ 

00 

+  00 

180 

1 

1.756 

I-75I 

1-743 

I-731 

1-715 

1.695 

1.671 

1.642 

1.607 

1.566 

179 

457 

455 

45° 

442 

43° 

414 

394 

3/0 

3 

41 

306 

265 

30.1 
_   £ 

178 

3 

279 

274 

266 

254 

238 

218 

194 

165 

130 

089 

17.6 

177 

4 

156 

154 

149 

141 

129 

113 

093 

069 

040 

005 

0.964 

12.5 

176 

5 

i.  060 

1.058  1.053 

1.045 

1-033 

1.017 

0.097 

0-973 

0.944 

0.909 

0.868 

9.6 

175 

6 

0.981 

0.979  i  0.974 

0.966 

0-954 

0.938 

c 

>i8 

894 

865 

8 

3° 

769 

7-9 

174 

7 

914 

912 

907 

899 

887 

871 

i 

51 

827 

798 

7 

63 

722 

6.7 

173 

8 

856 

854 

849 

841 

829 

813 

793 

769 

740 

705 

664 

5-8 

172 

9 

806 

804 

799 

791 

779 

763 

743 

719 

690 

655 

614 

5-° 

171 

10 

0.760 

0-759 

o-753 

0-745 

0-733 

0.717 

0.697 

0.673 

0.644 

0.6 

09 

0.568 

4.6 

17O 

11 

719 

717 

712 

704 

692 

676 

656 

632 

603 

5 

68 

527 

4.1 

169 

682 

680 

675 

667 

655 

639 

619 

595 

5 

66 

531 

490 

37 

168 

13 

648 

646 

641 

633 

621 

605 

585 

5 

61 

532 

497 

456 

3-4 

167 

14 

616 

614 

609 

60  1 

589 

573 

553 

529 

500 

465 

424 

3.2 

166 

15 

0.587 

0.585 

0.580 

0.572 

0.560 

0-544 

0.524 

0.500 

0.471 

0.436 

0.395 

2.9 

165 

16 

560 

558 

553 

545 

533 

517 

497 

473 

444 

409 

368 

2.6 

164 

17 

534 

S32 

527 

519 

507 

491 

t 

tfi 

447 

418 

383 

342 

163 

18 

510 

508 

5°3 

495 

483 

467 

447 

423 

394 

359 

3*8 

2.4 

162 

19 

488 

486 

481 

473 

461 

445 

425 

401 

372 

337 

296 

2.2 

161 

2.2 

2O 

0.466 

0.464 

0-459 

0.451 

0-439 

0.423 

0.403 

0-379 

0-350 

0-315 

0.274 

160 

21 

446 

444 

439 

43  i 

419 

403 

383 

359 

33° 

295 

254 

" 

159 

22 

427 

425 

420 

412 

400 

384 

364 

340 

3" 

276 

235 

1-9 

158 

23 

408 

406 

401 

393 

51 

365 

345 

321 

292 

257 

216 

1.9 

157 

24 

391 

389 

384 

376 

364 

348 

328 

304 

275 

240 

199 

*-7 

156 

25 

o-374 

0.372 

0.367 

0-359 

0.347 

0.331 

0.311 

0.287 

0.258 

0.223 

0.182 

r-7 
1.6 

155 

26 

358 

356 

3 

Si 

343 

31 

315 

295 

271 

242 

207 

1  66 

154 

27 

343 

336 

328 

316 

300 

280 

256 

227 

192 

151 

i«5 

153 

28 

328 

326 

321 

313 

301 

285 

265 

241 

212 

177 

136 

i>5 

152 

29 

3H 

312 

307 

299 

287 

271 

251 

227 

198 

163 

122 

1.4 

151 

30 

0.301 

0.299 

0.294 

0.286 

0.274 

0.258 

0.238 

0.214 

0.185 

0.150 

0.109 

i-3 

15O 

31 

288 

286 

281 

273 

261 

245 

225 

20  1 

172 

137 

096 

"3 

149 

32 

276 

274 

269 

261 

249 

233 

213 

i 

89 

1  60 

125 

084 

1.2 

148 

33 

264 

262 

257 

249 

237 

221 

20  1 

177 

148 

"3 

072 

' 

147 

34 

253 

251 

246 

238 

226 

210 

190 

166 

137 

102 

06  1 

LI 

146 

I.I 

35 

0.242 

0.240 

0-235 

0.227 

0.215 

0.199 

0.179 

O.I 

55 

0.126 

0.091 

0.050 

145 

36 

231 

229 

224 

216 

204 

1  88 

1  68 

144 

I 

15 

080 

039 

* 

144 

37 

221 

219 

214 

206 

194 

178 

158 

i 

34 

I05 

070 

029 

I.O 

143 

38 

211 

209 

204 

i 

96 

184 

168 

148 

124 

095 

060 

OI9 

I.O 

142 

39 

201 

199 

194 

i 

86 

174 

158 

138 

114 

085 

050 

009 

I.O 

141 

40 

0.192 

0.190 

0.185 

0.177 

0.165 

0.149 

0.129 

O.I 

0=5 

0.076 

0.041 

0.000 

0.9 

_  o.- 

140 

42 

175 

173 

i 

68 

160 

148 

132 

112 

o 

88 

059 

024 

9.983 

0.05 
0.8^ 

138 

44 

158 

156 

i 

51 

H3 

i 

31 

115 

095 

071 

042 

007 

966 

136 

46 

143 

141 

136 

i 

28 

116 

IOO 

080 

056 

027 

9.992 

951 

o-7S 

134 

48 

129 

127 

122 

114 

102 

086 

066 

042 

013 

978 

937 

0.70 

132 

0.65 

I  50 

0.116 

0.114 

0.109 

O.IOI 

0.089 

0.073 

0.053 

0.029 

0.000 

9.965 

9.924 

13O 

52 

104 

102 

O 

97 

089 

077 

06  1 

041 

017 

9.988 

953 

912 

128 

54 

092 

0 

§5 

077 

065 

049 

029 

005 

976 

941 

126 

56 

082 

075 

067 

°55 

039 

OI9 

9-9 

95 

966 

9 

3i 

0.50 

124 

58 

072 

070 

065 

057 

045 

029 

OO9 

9 

85 

956 

921 

0.50 

122 

60 

0.063 

0.06  1 

0.056 

0.048 

0.036 

O.O2O 

O.OOO 

9-976 

9-947 

9.9 

12 

9.871 

°'45 

12O 

65 

043 

041 

036 

028 

016 

000 

9.980 

956 

927 

8 

92 

851 

0.40 

115 

70 

027 

025 

O2O 

012 

ooo 

9.984 

964 

940 

ii 

876 

835 

0.32 

110 

75 

013 

008 

000 

9.988 

972 

952 

928 

£ 

99 

864 

823 

0.24 

105 

80 

007 

005 

000 

9.992 

980 

964 

944 

920 

891 

856 

815 

O.I 

100 

90 

o.ooo 

9-998 

9-993 

9.985 

9-973 

9-957 

9-937 

9.913 

9.884 

9.849 

9.808 

0.07 

90 

1 

0.04    o.xo 

0.16 

0.24 

0.40 

0.48 

0.58 

0.70 

0.82 

Az. 

I 

For 
Az. 

Az. 

Difference 

for  0°.l  of  Declination, 

154 


TABLE  XLVIII. 


Time  Alt 

•  Azimuths  :  Log  B. 

Hour-Angle 
or  Position- 

Altitude  or  Latitude. 

Hour-Angle 
or  Position- 

Angle. 

Angle. 

Arc. 

Time. 

O° 

10° 

15° 

20° 

25° 

30° 

32° 

34° 

34 

i°    3* 

|0 

4O° 

Time- 

Arc. 

o 

A     m 

k     m 

0 

0     0 

+  00 

_|_ 

oo 

_f_ 

oc 

_(- 

00 

_|_ 

00 

+    00 

_|_ 

00 

-j_ 

oo 

+ 

oo  + 

oc 

-j-    00 

12     0 

180 

1 

0     4 

1-758 

1.7 

51 

1-743 

1.7 

31 

1.715 

1.695 

1.686 

1.676 

1.666  1.654 

1.642 

11  56 

179 

2 

0     8 

457 

450 

442 

43° 

414 

394 

385 

375 

365 

353 

341 

11  52 

178 

3 

O  12 

281 

274 

266 

254 

238 

218 

209 

199 

i 

»9 

177 

165 

11  48 

177 

4 

0  16 

156 

149 

141 

129 

"3 

093 

084 

074 

064 

052 

040 

11  44 

176 

5 

020 

i.  060 

1-053 

1.045 

I-°33 

1.017 

0-997 

0.9 

SS 

0.978 

0.968 

0.956 

0.944 

11  40 

175 

6 

0  24 

0.981 

0-974 

0.966 

o.954 

0.9 

^8 

918 

909 

899 

8 

89 

8 

77 

865 

11  36 

174 

7 

O  28 

914 

9 

37 

899 

887 

8 

7i 

851 

8 

42 

832 

822 

810 

798 

11  32 

173 

8 

O  32 

856 

8 

49 

841 

829 

813 

793 

7 

84 

774 

764 

752 

740 

11  28 

172 

9 

0  36 

806 

799 

791 

779 

763 

743 

734 

724 

7H 

702 

690 

11  24 

171 

10 

040 

0.760 

o-753 

0-745 

o-733 

0.717 

0.697 

0.688 

0.678 

0.668 

0.656 

0.644 

11  2O 

170 

11 

044 

719 

712 

704 

692 

676 

656 

647 

637 

627 

6 

15 

603 

11  16 

169 

12 

O  48 

682 

675 

667 

655 

639 

619 

610 

600 

590 

578 

566 

11  12 

168 

13 

0  52 

648 

641 

633 

621 

605 

585 

576 

566 

556 

544 

532 

11      8 

167 

14 

0  56 

616 

609 

601 

589 

573 

553 

544 

534 

524 

512 

500 

11     4 

166 

15 

0 

0.587 

0.580 

0-572 

0.560 

0-544 

0.524 

0.515 

0-505 

0-495 

0.483 

0.471 

11     O 

165 

16 

4 

560 

553 

545 

533 

5 

17 

497 

488 

478 

468 

456 

444 

10  56 

164  | 

17 

8 

534 

527 

5 

10 

5°7 

491 

462 

452 

442 

43° 

418 

10  5-2 

163 

18 

12 

5*0 

5°3 

495 

483 

467 

447 

438 

428 

418 

406 

394 

1O  48 

162 

19 

16 

488 

481 

473 

461 

445 

425 

416 

406 

396 

384 

372 

10  44 

161 

29 

20 

0.466 

o.459 

0-45  * 

0-439 

0.423 

0.403 

0-394 

0.384 

0-374 

0.362 

0-350 

10  40 

16O 

21 

24 

446 

439 

43  1 

419 

403 

383 

374 

364 

354 

342 

330 

10  36 

159 

22 

28 

427 

420 

412 

400 

384 

364 

355 

345 

335 

323 

3" 

1O  32 

158 

23 

32 

408 

401 

393 

3 

Si 

365 

345 

336 

326 

316 

3°4 

292 

1O  28 

157 

24 

36 

39  * 

384 

3 

76 

364 

3 

48 

328 

3 

19 

3°9 

299 

287 

275 

10  21 

156 

25 

!  26 

40 
44 

o-374 
358 

0.367 

35  i 

o-359 
343 

0-347 
33  i 

o.33i 

0.311 

295 

0.302 
286 

0.292 
276 

0.282 
266 

0.270 
254 

0.258 
242 

10  20 
10  16 

155 
154 

27 

48 

343 

336 

328 

316 

300 

280 

271 

261 

251 

239 

227 

10  12  i  153 

28 

52 

328 

321 

313 

301 

285 

265 

256 

246 

236 

224 

212 

1O     8     152 

29 

56 

3*4 

307 

299 

287 

271 

251 

242 

23? 

222 

210 

I98 

10     4 

151 

30 

2     0 

0.701 

0.294 

0.286 

0.274 

0.258 

0.238 

0.229 

0.219 

0.209 

O.I 

97 

0.185 

1O     O 

15t> 

31 

2     4 

288 

281 

273 

261 

245 

225 

216 

206 

I 

96 

I 

84 

172 

9  56 

149 

32 

2     8 

276 

269 

261 

249 

233 

213 

204 

194 

I 

84 

172 

1  60 

9  52 

148  ; 

33 

2  12 

264 

257 

249 

237 

221 

20  1 

i 

92 

182 

172 

1  60 

148 

9  48 

147 

34 

2  16 

253 

246 

238 

226 

2IO 

190 

i 

81 

171 

161 

149 

137 

9  44 

146 

35 

2  20 

O.242 

0-235 

0.227 

0.215 

0.199 

0.179 

0.170 

o.i  60 

0.150 

O.I 

38 

0.126 

9  4O 

145 

36 

2  24 

231 

224 

216 

204 

1  88 

1  68 

'59 

149 

139 

127 

H5 

9  36 

144 

37 

2  28 

221 

214 

206 

194 

178 

158 

149 

139 

129 

117 

105 

9  32    143 

!  38 

2  32 

211 

204 

i 

96 

184 

1  68 

148 

139 

129 

119 

107 

095 

9  28    142 

39 

2  36 

2O  I 

194 

i 

86 

174 

158 

138 

129 

119 

109 

097 

085 

9  24 

141 

40 

240 

O.I92 

0.185 

0.177 

0.165 

0.149 

0.129 

0.120 

O.IIO 

O.I  00 

0.088 

0.076 

9  2O 

140 

42 

2  48 

175 

168 

160 

148 

132 

112 

103 

093 

083 

071 

059 

9  12    138 

44 

2  56 

158 

i 

5i 

143 

'31 

"5 

095 

O 

86 

076 

066 

054 

042 

9     4    136 

46 

3     4 

136 

i 

28 

116 

IOO 

080 

071 

06  1 

05  I 

039 

027 

8  56 

134 

48 

3  12 

129 

122 

114 

1  02 

086 

066 

•057 

047 

037 

025 

013 

8  48 

132 

5O 

3  20 

0.116 

O.IO9 

O.IOI 

0.089 

0.073 

0-°53 

0.044 

0.034 

0.024 

0.012 

O.OOO 

8  40 

130 

52 

3  28 

104 

O 

97 

089 

077 

061 

041 

032 

O2  2 

012 

OOO 

9.988 

8  32 

128 

54 

3  36 

092 

0 

85 

077 

065 

049 

029 

O2O 

010 

OOO 

9.988 

976 

8  24 

126 

56 

3  44 

082 

°75 

067 

°55 

039 

019 

010 

ooo  9.990 

978 

966 

8  16 

124 

58 

3  52 

072 

065 

057 

045 

029 

O09 

OOO 

9.990  i  980 

9 

68 

956 

8     8 

122 

60 

4     0 

0.063 

0.056 

0.048 

0.036 

0.020 

0.000 

9.991 

9.981  9.971 

9-959 

9-947 

8     0 

12O 

65 

4  20 

043 

036 

028 

016 

OOO 

9.980 

971 

961 

951 

939 

927 

7  40 

115 

70 

4  40 

027 

O2O 

012 

ooo 

9-9 

84 

964 

955 

945 

935 

923 

911 

7  2O 

no 

75 

5     0 

015 

008 

000 

9.988 

972 

952 

943 

933 

923 

911 

899 

7     O 

105 

80 

5  20 

007 

OOO 

9.992 

980 

964 

944 

935 

925 

915 

903 

891 

6  40 

1OO 

90 

6     0 

o.ooo  9.993 

9-985  9-973 

9-957 

9-937 

9.928  9.918 

9.908  9.896 

9.884 

6     O 

90 

0.07 

0.16 

024 

0.32 

0.40   1   045 

0.50 

o  50 

0.60 

0.60 

Differences 

for  0°.l  of  Altitude  or  Latitude. 

Differences  for  Side-  Arguments 

are  the  same  as 

for  Azimuths  under  Log  A. 

TABLE  XLVIII. 


155 


Time-Alt.  Azimuths:  L,og 

B. 

Hour-Angle 

Altitude  or  Latitude. 

Hour-Angle 

or  Position 

or  Position- 

Angle. 

Angle. 

Arc. 

Time. 

40D 

42° 

44° 

46° 

48° 

50° 

51° 

52° 

53° 

54° 

55° 

Time. 

Arc. 

o 

h      m 

h       m 

o 

0 

0     0 

-f-  oo 

4  oo 

4  oo 

4  oo 

4  oo 

4  oo 

4  °° 

4  oo 

4  oo 

4  oo 

4  oo 

12     0 

180 

1 

0     4 

1.642 

1.629 

1.615 

i.  600 

I. 

583 

1.566 

i-557 

1-547 

1-537 

1.527 

1.517 

11  56 

179 

2 

0     8 

341 

328 

3H 

299 

282 

265 

256 

246 

236 

226 

216 

11  52 

178 

3 

0  12 

165 

152 

138 

123 

1  06 

089 

080 

070 

060 

050 

040 

11  48 

177 

4 

O  16 

040 

027 

013 

0.998 

0.981 

0.964 

0-955 

0-945 

0-935 

0.925 

0.915 

11  44 

176 

5 

0  20 

0.944 

0.931 

o. 

}i7 

o. 

902 

0.885 

0.868 

0.859 

0.849 

0.839 

0.829 

0.819 

11  40 

175 

6 

O  24 

865 

852 

i 

3?8 

^23 

806 

789 

780 

770 

760 

740 

11  36 

174 

7 

O  28 

798 

7*5 

771 

756 

739 

722 

703 

693 

683 

673 

11  32 

173 

8 

0  32 

740 

727 

713 

< 

598 

58i 

664 

655 

645 

635 

625 

615 

11  28 

172 

9 

0  36 

690 

677 

663 

648 

631 

614 

605 

595 

585 

575 

565 

11  24 

171 

10 

040 

0.644 

0.631 

0.617 

0.602 

0.585 

0.568 

0-559 

0-549 

0-539 

0.529 

0.519 

11  20 

170 

11 

O  44 

603 

590 

576 

561 

544 

527 

518 

508 

498 

488 

478 

11  16 

169 

12 

048 

566 

553 

539 

524 

507 

490 

481 

< 

tfi 

46! 

i 

I5i 

441 

11  12 

168 

13 

O  52 

532 

t 

;?9 

505 

490 

473 

456 

447 

437 

427 

i 

407 

11     8 

167 

14 

0  56 

500 

487 

473 

458 

441 

424 

415 

405 

395 

385 

375 

11     4 

166 

15 

O 

0.471 

0.458 

0.444 

0.429 

0.412 

o-395 

0.386 

0.376 

0.366 

0.356 

0.346 

11     0 

165 

16 

4 

444 

43i 

417 

402 

385 

368 

359 

349 

339 

329 

319 

10  56 

164 

17 

8 

418 

405 

39i 

376 

369 

342 

333 

323 

,'3 

303 

293 

10  52 

163 

18 

12 

394 

381 

367 

352 

335 

309 

299 

289 

279 

269 

1048 

162 

19 

16 

372 

359 

345 

330 

513 

196 

287 

277 

267 

257 

247 

1044 

161 

2O 

20 

0.350 

0.337 

0-323 

0.308 

0.291 

0.274 

0.265 

0-255 

0.245 

0-235 

0.225 

1040 

16O 

21 

24 

33° 

317 

3°3 

288 

271 

254 

245 

235 

225 

215 

205 

1O  36 

159 

22 

28 

311 

298 

284 

269 

252 

235 

226 

216 

206 

196 

186 

10  32 

158 

23 

32 

292 

279 

265 

250 

233 

216 

207 

197 

187 

177 

167 

10  28 

157 

24 

36 

275 

262 

248 

233 

216 

199 

190 

80 

170 

1  60 

150 

10  24 

156 

25 

40 

0.258 

0.245 

0.231 

0.216 

0.199 

0.182 

0.173 

0.163 

0.153 

0-143 

0.133 

1O  2O 

155 

26 

44 

242 

229 

215 

200 

183 

166 

147 

J37 

127 

117 

10  16 

154 

27 

48 

227 

214 

200 

I85 

68 

I51 

142 

132 

122 

112 

102 

10  12 

153 

28 

52 

212 

i 

99 

185 

170 

] 

53 

136 

127 

117 

107 

C 

,97 

087 

10     8 

152 

29 

56 

198 

i 

85 

171 

I56 

149 

122 

"3 

103 

093 

C 

83 

073 

10     4 

151 

30 

2     0 

0.185 

0.172 

0.158 

0.143 

0.126 

0.109 

O.I  00 

0.090 

0.080 

0.070 

0.060 

10     0 

150 

31 

2     4 

172 

159 

i 

45 

130 

"3 

096 

087 

077 

067 

057 

047 

9  56 

149 

32 

2     8 

1  60 

147 

133 

n8 

101 

084 

075 

065 

055 

045 

°35 

9  52 

148 

33 

2  12 

148 

i 

35 

121 

1  06 

089 

072 

063 

°53 

043 

033 

023 

9  48 

147 

34 

2  16 

137 

124 

no 

095 

078 

06  1 

052 

042 

032 

O22 

012 

944 

146 

35 

2  2O 

0.126 

0.113  0.099 

0.084 

0.067 

0.050 

0.041 

0.031 

0.021 

O.OII 

O.OOO 

9  40 

145 

36 

2  24 

"5 

102 

088 

073 

056 

039 

036 

'  020 

OIO 

000 

9.990 

9  36 

144 

37 

2  28 

105 

C 

92 

078 

063 

046 

029 

O2O 

OIO 

OOO 

9.990 

980 

9  32 

143 

38 

2  32 

095 

c 

82 

068 

053 

036 

OI9 

OIO 

ooo 

9-9 

9° 

980 

970 

9  28 

142 

39 

2  36 

085 

072 

058 

043 

026 

009 

000 

9.990 

9 

80 

970 

960 

9  24 

141 

40 

2  40 

0.076 

0.063 

0.049 

0.034 

0.017 

0.000 

9.991 

9.98! 

9.971  9.961 

9-951 

9  20 

14O 

42 

248 

059 

046 

032 

017 

ooo 

9.983 

974 

964     954 

944 

934 

9  12 

138 

44 

2  56 

042 

029 

OI5 

000 

9-C 

•83 

966 

957 

947 

937 

927 

917 

9     4 

136 

46 

3     4 

027 

014 

000 

9-985 

968 

951 

942 

932 

922 

9 

12 

8  56 

134 

48 

3  12 

013 

000 

9.986 

971 

954 

937 

928 

918 

908 

S 

98 

888 

8  48 

132 

50 

3  20 

o.ooo 

9.987 

9-973  9.9S8 

9.941 

9.924 

9-9I5 

9-9 

°5 

9.8 

95 

9.885 

9-875 

8  40 

130 

52 

3  28 

9.988 

975 

961 

946 

939 

912 

903 

g 

93 

8 

8} 

873 

863 

8  32 

128 

54 

3  36 

976 

963 

949 

934 

917 

900 

891 

881 

871 

861 

851 

8  24 

126 

56 

3  44 

966 

953 

939 

924 

907 

890 

88  1 

871 

861 

851 

841 

8  16 

124 

58 

3  52 

956 

943 

929 

914 

897 

880 

871 

86  1 

851 

841 

831 

8     8 

122 

60 
65 

4     0 
4  20 

9.947 
927 

9-934  J9-92Q  19.905 
r4     900!    88q 

9.888 
868 

9.871 
851 

9.862 
842 

9.852 
832 

9.842 

822 

9.832 
812 

9.822 
802 

8     0 
7  40 

120 
115 

70 

440 

911 

98 

8 

84     8 

69 

852 

835 

826 

816 

806 

96 

786 

7  20 

110 

75 

5     0 

899 

886 

872     8^7 

840 

823 

814 

804 

7 

Q4 

. 

§4 

774 

7     0 

105 

80 

5  20 

891 

878 

864 

849 

832 

815 

806 

796 

7 

S6 

776 

766 

640 

100 

90 

6     0 

9.884  9.871 

9.857  9.842 

9.825 

9.808 

•799  9-789  9-779  9-769 

9-759 

6     0 

90 

0.65 

0.70 

o-75 

0.85 

0.85    1     0.9     1    i.o 

I.O 

I.O 

I.O 

Differences 

for  0°-1  of  Altitude  or  Latitude. 

Differences  for  Side-  Arguments 

are  the  same  as 

for  Azimuths  under  Log  A. 

156 


TABLE  XL VIII. 


Time-  A  It 

.  Azimuths 

:  Log 

B. 

Hour-Angle 
or  Position- 

Altitude  or  Latitude. 

Hour-Angle 
or  Position- 

Angle. 

Angle. 

Ar;. 

Time. 

55° 

56° 

57°   58° 

59° 

60° 

61° 

62° 

63° 

64° 

65° 

Time. 

Arc. 

o 

h     in 

h        m 

0 

0     0 

+    00 

-\- 

00    + 

00    -f 

00 

-}- 

oo 

4-  oo 

+    00 

+    00 

_i_ 

oo 

_|_ 

oo 

4-    00 

12     0 

180 

1 

2 

0     4 

0     8 

1.517 

216 

1.506 
205 

1.494:1.482 
193!    181 

1.470 
169 

1-457 
156 

1.444  1.430  1.415 

143      1291    H4 

1.400 
099 

1.384 

083 

11  56 
11  52 

179  ! 
178  i 

3 

O  12 

040 

029 

01  7  j    005 

O.Q 

93 

0.980 

0.967 

0-953  0.938 

0.923 

0.907 

11  48 

177 

4 

0  16 

0.915 

0.904 

0.892  !o.88o 

8 

bS 

855 

842 

828.    813 

798 

782 

11  44 

176 

5 

0  20 

0.819 

0.808  0.796  0.784 

0.772 

0-759 

0.746 

0.732  0.717 

0.702 

0.686 

11  40 

175 

6 

0  24 

740 

729     717     705 

693 

680 

667 

653 

638 

623 

607 

11  36 

174 

7 

0  28 

673 

662 

650     638 

626 

613 

600 

586J   5 

71 

556 

54° 

11  32 

173  i 

8 

0  32 

615 

604 

592     580 

568 

555 

542 

528     513 

498 

482 

11  28 

172 

9 

0  36 

565 

554  1   542     53° 

518 

505 

492 

478 

463 

448 

432 

11  24 

171 

1O 

040 

0.519 

0.508  10.496  '0.484  0.472 

0-459 

0.446 

0.432 

0.4I7J0.402 

0.386 

li  20 

170 

11 

O  44 

478 

467 

455:    443|   431 

418 

405 

39i 

376     361 

345 

11   16 

169 

I  j_^ 

O  48 

441 

43° 

418     406 

394 

381 

368 

354  i    339     324      3°8 

11  12 

168 

13      O  52 

407 

396 

384     372 

360 

347 

334 

320!    305     290      274 

11      8 

167 

14 

0  56 

375 

364 

352     340 

328 

302 

2881    273     258 

242 

11      4 

166 

15 

o 

0.346 

0-335 

0.323  0.311 

0.299 

0.286 

0.273 

0.259  0.244  10.229 

0.213 

11      0 

165 

16 

319 

308 

296 

284 

272 

259 

246 

232  1    217 

202  j     1  86 

10  56 

164 

17 

8 

293 

282 

270 

258 

246 

233 

220 

206     191 

176      160 

1O  52 

163 

I  18 

12 

269 

258 

246 

234 

222 

209 

196 

182     167 

152      136 

1O  48 

164 

I  19 

16 

247 

236 

224       212 

2OO 

187 

174 

160  '    145      i 

30       114 

10  44 

161 

!  20 

20 

0.225 

0.214  0.202   0.190 

0.178 

0.165 

0.152 

0.138  0.123  °-I08  0.092 

1O  4O 

169 

21 

24 

205 

194!     l82j     I 

/o 

I58 

145 

132 

ii8j    103     088  !    072 

10  36 

159 

22 

28 

186 

175        l63J     151 

139 

126 

113 

099     084  1    069  i    053 

1O  34 

158 

23 

32 

167 

156       144       132 

1  2O 

107 

094 

080  1    065     050      034 

1O  28 

157 

24 

36 

150 

139 

127 

"5 

I03 

090 

077 

063  !   048  i   033  i    017 

10  24     a  .*« 

i  25          4O 

0.133 

O.I22  O.IIO  0.098 

0.086 

0.073 

0.060 

0.046 

0.031  o.o  1  6  j  o.ooo 

1O  2O    155 

!  26 

44 

117 

1  06 

094       082 

070 

057 

044 

030 

015     ooo  9.984 

14>  16  !  154 

27 

48 

1  02 

091     079  !    067 

055 

042 

029 

015 

ooo     985 

969 

1O  12  i  153 

28 

52 

087 

076 

064  i     052 

040 

027 

014  i    ooo 

9.985  9.970 

954 

10     8  i  152 

29 

56 

073 

062 

050       038 

O26 

013 

ooo  9.986     971 

956      940 

1O     4    151 

30 

2     O 

0.060 

0.049  0.037  0.025 

O.OI3 

0.000 

9-987  9-973  9.958 

9-943   9-927 

10     0 

150 

31 

2     4 

047 

036 

024;     OI2 

OOO 

9.987 

974 

960 

945 

930      914 

9  56 

149 

32 

2     8 

035 

024 

OI2  !     OOO 

9.988 

975 

962 

948 

933 

918 

902 

9  52 

148  j 

|33 

2  12 

023 

012 

ooo  19.988 

976 

963 

95o|    936 

921 

906 

890 

9  4§ 

147 

|  34 

2  16 

012 

OOO 

9.989     977 

965 

952 

939     925 

909 

895 

879 

9  44 

146 

35 

2  20 

O.OOO 

9.990 

9.978  9.966. 

9.954 

9.941 

9.928  9.914 

9.899 

9.884 

9.868 

9  40 

145 

36 

2  24 

9.990 

979 

967 

955 

943 

93° 

917 

903 

888 

873 

857 

9  36 

144 

37 

2  28 

980 

969 

957 

945 

933 

920 

878 

863 

847 

9  32 

143 

38 

2  32 

970 

959 

947 

935 

923 

910 

§7 

883 

868 

853 

837 

9  28 

142 

39 

2  36 

960 

949 

937 

925 

913 

900 

7 

873 

858 

843 

827 

9  24 

141 

40 

240 

9-95  ! 

9.940 

9.928 

9.916 

9-904 

9.891 

9.878 

9-864 

9.849 

34 

9.818 

9  20 

140 

42 

2  48 

934 

923 

9 

ii 

899 

887 

874 

86  1 

847 

832 

8i7 

80  1 

9  12 

138 

44 

2  56 

917 

9 

06 

8 

94 

882 

870 

857 

844 

830 

815 

800 

784 

9     4 

136 

46 

3     4 

902 

891 

879 

867 

855 

842 

829 

815 

800 

785 

8  56 

134 

48 

3  12 

888 

877 

865 

853 

841 

828 

815 

801 

786 

8  48 

132 

5O 

3  20 

9.875 

9.864 

9-852 

9.840 

9.828 

9.815 

9.802 

9788 

8  4O 

130 

52 

3  28 

863 

852 

840 

828 

816 

803 

790 

8  32 

128    \ 

54 

3  36 

851 

840 

828 

816 

804 

791 

8  24 

126 

56 

3  44 

841 

830 

818 

806 

7Q4 

8  16 

124 

58 

3  52 

831 

820 

808 

796 

8     8 

122 

60 

4     0 

9.822 

9.811 

9-799 

8     0 

120 

65 

4  20 

802 

791 

7  4O 

115 

70 

440 

7  20 

110 

75 

5     0 

7     0 

105 

!  80 

5  20 

6  4O 

100  | 

90 

6     0 

6     0 

90  ! 

1 

x 

2 

1.3    1    1.3   1   1.4   1  1.5 

i-S 

1.6 

Differences 

for  0°.l  of  Altitude  or  Latitude. 

Differences  for  Side-  Arguments  are  the  same  as  those  for  Azimuths  under  Log  A. 

TABLE  XLVIII. 


157 


Time-Alt.  Azimuths:  Log  B. 

Hour-Angle 
or  Position- 

Altitude  or  Latitude. 

Hour-Angle 
or  Position- 

Angle. 

Angle. 

Arc. 

Time. 

65° 

66° 

67° 

68° 

69° 

70° 

71° 

72° 

73° 

74° 

75° 

Time. 

Arc.    \ 

o 

h      m 

h       m 

0 

0 

0     0 

+    00 

+    00 

+ 

oo 

+ 

00   + 

00 

+    00 

+    00 

+ 

oo 

+    00 

+ 

00 

+    05 

12     0 

18O 

1 

0     4 

1.384 

1.367 

I-35° 

1.331  1.312 

1.292 

1.270 

1.2 

48 

1.224 

1.198 

1.171 

11  56 

179 

2 

0     8 

083 

066 

049 

030     01  1 

0.991 

0.979 

0.947 

0.923 

0.897 

0.870 

11  52 

178 

3 
4 

0  12 
O  16 

0.907 

782 

0.890 

765 

0.873 
748 

0.854  0.835 
729     710 

815 

690 

1^ 

771 
636 

747 
622 

721 

596 

694 
569 

11  48 
11  44 

177 
176 

5 

0  20 

0.686 

0.669 

0.652 

0.633  0.614 

0-594 

0.572 

°-5 

5° 

0.526 

0.500 

0-473 

11  40 

175 

6 

O  24 

607 

590 

573 

5 

54 

5 

35 

493 

47i 

447 

421 

394 

11  36 

174 

7 

O  28 

540 

523 

506 

4 

«7 

4 

68 

448 

426 

404 

380 

3 

54 

327 

11  32 

173 

8 

0  32 

482 

465 

448 

429 

410 

390 

346 

322 

2 

?6 

269 

11  28 

172 

9 

0  36 

432 

415 

398 

379 

360 

34° 

3i8 

296 

272 

246 

219 

11  24 

171 

10 

O  4O 

0.386 

0.369 

0.352 

0-333  0-3i4 

0.294 

0.272  0.250 

0.226 

O.2OO 

0.173 

11  20 

170 

11 

044 

345 

328 

3 

II 

292  1    273 

253 

231 

209 

185 

159 

132 

11  16 

169 

12 

O  48 

308 

291 

274 

255 

236 

216 

194 

172 

148 

122 

095 

11  12 

168 

13 

O  52 

274 

257 

240 

221 

202 

182 

1  60 

138 

114 

088 

061 

11      8 

167 

14 

0  56 

242 

225 

208 

189       1  70 

150 

128 

1  06 

082 

056 

029 

11     4 

166 

15 

O 

0.213 

0.196 

0.179 

0.160  0.141 

O.I2I 

0.099 

0.077 

0-053 

O.O27 

O.OOO 

11     0 

165 

16 

4 

1  86 

169 

i 

52 

I 

33 

114 

094 

072 

050 

026 

OOO 

9-973 

10  56 

164 

17 

8 

1  60 

126 

107       088 

068 

046 

024 

000 

9-974 

947 

10  52 

163 

18 

12 

136 

119 

102 

083       064 

044 

022 

ooo 

9.976 

95° 

923 

10  48 

162 

19 

16 

114 

097 

080 

O6  1       042 

022 

000 

9.978 

954 

928 

901 

10  44 

161 

2t> 

20 

0.092 

0.075 

0.058 

0.039  :o.o2o 

O.OOO 

9.978 

9-956 

9-932 

9-9 

36 

9.879 

10  40 

160 

21 

21 

072 

055 

O 

}8 

019     ooo 

9.980 

958 

936 

912 

886 

859 

1.0  36 

159 

22 

28 

053 

036 

019 

ooo  9.981 

961 

949 

i7 

893 

867 

840 

10  32 

158 

23 

32 

034 

017 

ooo  9.981     962 

942 

920 

£ 

98 

874 

8 

iB 

821 

10  28 

157 

24 

36 

017 

000 

9.983     964     945 

925 

9°3 

881 

857 

831 

804 

10  24 

156 

|  25 

4O 

0.000 

9-983 

9.966 

9.947  9.928 

9.908 

9.886 

9.864 

9.840 

9.814 

9787 

1O  2O 

155 

26 

44 

984 

967 

9 

5° 

931  !    912 

892 

870 

8 

48 

824 

7 

58 

10  16 

154 

27 

48 

969 

952 

9 

916  i   8 

97 

877 

855 

833 

809 

7 

S} 

10  12 

153 

28 

52 

954 

937 

920 

9 

91 

c> 
O 

82 

862 

840 

818 

794 

10     8 

152 

29 

56 

940 

923 

906 

8 

87 

868 

848 

826 

804 

1O     4 

151 

30 

2     0 

9.927 

9.910 

9.893 

9.874 

9-855 

9.835 

9.813 

9.791 

10     0 

150 

!  31 

2     4 

914 

897 

880 

861 

842 

822 

800 

9  56 

149 

32 

2     8 

902 

885 

868 

849 

830 

810 

788 

9  52 

148 

33 

2  12 

890 

873 

856 

837 

818 

798 

9  48 

147 

34 

2  16 

879 

862 

845 

826 

807 

787 

9  44 

146 

35 

2  2O 

9.868 

9.851 

9-834 

9.815  9.796 

9  4O 

145 

36 

2  24 

857 

840 

823 

804 

9  36 

144 

37 

2  28 

847 

830 

*3 

794 

9  32 

143 

38 

2  32 

837 

820 

8 

03 

9  28 

142 

39 

2  36 

827 

810 

793 

9  24 

141 

40 

2  40 

9.818 

9.801 

9  20 

140 

42 

2  48 

801 

9  12 

138 

44 

2  56 

9     4 

136 

46 

3     4 

8  56 

134 

48 

3  12 

8  48 

132 

50 

3  2O 

8  40 

130 

52 

3  28 

8  32 

128 

54 

3  36 

8  24 

126 

56 

3  44 

8  16 

124 

58 

3  52 

8     8 

122 

60 

4     0 

8     0 

120 

65 

4  20 

7  40 

115 

70 

440 

7  2O 

110 

75 

5     0 

7     0 

105 

80 

5  20 

6  40 

1OO 

90 

6     0 

6     O 

90 

1.7   1   1.7 

1.9 

1.9 

2.0       1        2.2      j      2.2 

2.4         2.6 

2.7 

Differences 

for  0°.l  of  Altitude  or  Latitude. 

Differences  for  Side-  Arguments  are  the  same  as  those  for  Azimuths  under  Log  A. 

158 


TABLE  XLIX. 


Error  of  Time-  Alt.  Azimuth  for  an  Error  of  Om.2,  or  12%  in  Hour-Angle. 

Hour-  Angle. 

Az. 

i 

0  0 

Az. 

6  0 

5  20 

4  40 

4  O  !  3  20    2  40 

2  0 

1  40 

1  20 

100  40 

0  20 

1 

0 

o 
O.O 

o' 

o' 

o' 

o' 

o' 

o' 

O.O 

o 
0.0 

c 
O.O 

O.O 

o 
0.0 

Indet, 

180 

IO 

O 

o 

0 

i 

I 

I 

i 

o 

o 

0 

I 

I 

00 

170 

20 

0 

o 

I 

I 

I 

I 

2 

o 

I 

I 

I 

2 

oo 

16O 

30 

o 

I 

i 

2 

2 

3 

I 

I 

I 

2 

3 

00 

15O 

40 

o 

1 

2 

2 

3 

4 

I 

1 

2 

2 

5 

00 

140 

50 

O.O 

I 

2 

3 

4 

6 

O.I 

0.2 

0.2 

0.3 

0.7 

oo 

13O 

60 

o 

2 

3 

4 

6 

9 

2 

2 

3 

5 

I.O 

00 

12O 

65 

o 

2 

4 

5 

8 

ii 

2 

3 

4 

6 

2 

00 

115 

70 

0 

3 

5 

7 

10 

14 

3 

4 

5 

8 

6 

00 

no 

75 

o 

2 

4 

7 

9 

13 

19 

4 

5 

7 

i.i 

2.1 

00 

1O5 

8O 

0.0 

3 

6 

IO 

I5 

20 

29 

0.6 

0.8 

i.i 

1.6 

3-2 

00 

100 

81 

o 

3 

7 

ii 

16 

23 

33 

7 

9 

2 

8 

6 

00 

99 

82 

0 

4 

8 

12 

18 

25 

37 

8 

I.O 

3 

2.0 

4.1 

00 

98 

83 

0 

4 

9 

H 

21 

29 

42 

9 

i 

5 

3 

6 

00 

97 

84 

o 

5 

10 

17 

24 

34 

49 

I.O 

3 

8 

7 

5-4 

oc 

96 

85 

O.O 

6 

I3 

20 

29 

4i 

60 

1.2 

1.6 

2.1 

3-2 

6.s 

95 

86 

o        8 

It) 

2S 

51 

75 

5 

2.O 

7 

4.0 

94 

87 

O        IO 

21 

33 

48 

69 

99 

2.0 

6 

3.6 

93 

j  88 
89 

o      15 

o      30 

is 

99 

72 
144 

103 
205 

149 
297 

6J 

3-9 

92 
91 

9O 

Iiidet. 

00 

00 

oo 

00 

00 

00 

00 

90 

6  0 

6  40 

7  20 

8  0 

8  40 

9  20 

10  0 

1020 

1040 

11  0 

1120 

1140 

12  0 

Az 

Az. 

Hour-Angle. 

TABLE  L. 

Error  of  Time-  Alt.  Azimuth  for  an  Error  of  3'  in  Altitude  or  Declination. 

Altitude  or  Declination. 

Az 

Az. 

0° 

10° 

2O° 

30° 

40° 

5O° 

60° 

65° 

7O° 

75° 

80° 

85° 

90° 

0 

o 

o' 

o' 

o' 

o' 

o' 

o' 

o' 

0 

O.O 

o 
0.0 

o 
0.0 

O.O 

o 
0.0 

Indet. 

O     | 

180 

IO 

o 

o 

0 

i 

i 

i 

i 

0 

0 

O 

I 

I 

00 

17O 

2O 

0 

0 

I 

i 

I 

i 

2 

o 

I 

I 

I 

2 

oo 

16O 

3O 

o 

I 

i 

2 

2 

3 

I 

I 

I 

2 

3 

00 

15O 

40 

0 

I 

2 

2 

3 

4 

I 

I 

2 

2 

5 

00 

14O 

5O 

0 

I 

2 

3 

4 

6 

O.I 

0.2 

0.2 

0-3 

0.7 

oo 

130 

60 

0 

2 

3 

4 

6 

9 

2 

2 

3 

I.O 

00 

12O 

65 

o 

2 

4 

5 

8 

ii 

2 

3 

4 

6 

2 

00 

115 

7O 
75 

0 
0 

2 

3 

4 

5 
7 

7 
9 

10 

13 

H 
19 

3 

4 

4 
5 

5 
7 

8 
i.i 

6 

2.1 

00 
00 

no 

105 

80 

0 

3 

6 

10 

I5 

20 

29 

0.6 

0.8 

i.i 

1.6 

3-2 

oo 

100  | 

81 

82 

o 

0 

3 
4 

2 

ii 

12 

16 

18 

23 
25 

33 
37 

I 

9 

I.O 

2 

3 

8 

2.O 

6 
4.1 

00 

oc 

99 

98  ; 

83 

0 

4 

9 

14 

21 

29 

42 

9 

i 

3 

6 

00 

97  ! 

84 

o 

5 

10 

17 

24 

34 

49 

I.O 

3 

8 

7 

5-4 

00 

96  - 

85 

0 

6 

13 

20 

29 

41 

60 

1.2 

1.6 

2.1 

3-2 

6.5 

95  ! 

86 

87 

o 
o 

8 

IO 

1  6 

21 

25 

.36 
'48 

69 

75 
99 

5 

2.0 

2.0 

6 

3! 

4.0 

94  1 
93 

88 
89 

0 
0 

'5 

30 

32 

63 

99 

72 
144 

103 
205 

149 
297 

6.? 

3-9 

92 
91 

90 

Indet 

00 

oo 

00 

00 

00 

00 

90 

TABLE  LI. 


159 


Transition-  Azimuths. 

T    _4 

Change  of  Altituds  in  One  Minute  of  Time. 

Li  at. 

01        2 

3 

4' 

5 

6 

7'       8' 

9 

10 

11     12' 

13 

14' 

15'' 

o 

o 

o            o 

o 

o 

o 

o 

o 

o 

0 

0 

o             o 

o 

o 

o 

O 

o.o 

3-8      7-7 

"•5 

15-5 

19-5 

23.6 

27.8 

32.2 

36-9 

41.8 

47-2 

53-1 

60.  1 

69.0 

90.0  1 

4 

o  !  3.8    7.7 

II.6 

15-5 

19.5 

23.61  27.9 

32.3  1  37.0 

41.9 

47-3 

53-3 

60.3 

69.3 

§ 

o       3-9 

7-7 

II.  7 

15.6 

19.7 

23.8   28.1 

32-6}  37-3 

42.3 

47-8 

53-9 

61.1 

70.5 

12 

o       3.9!     7.8  |  1  1.8 

15.8 

19.9 

24.1   28.5 

33-o  |  37-8 

43-o 

48.6 

54-9 

62.4    72.6 

16 

o       4.0  1    8.0 

I2.O 

16.1 

20.3 

24.6;  29.0  j  33.7  38.6 

43-9 

49-7    56-3 

64.4!  76-I 

20 

o.o  |    4.1      8.2 

12.3 

16.5 

20.8 

25.2  i  29.8!  34.6  39.7 

45-2 

5i-3    58-3 

67-3 

83.3 

22 

o       4.1      8.3     12.4 

16.7 

21.  1 

25-5i  3°-2:  35-1 

40.3 

46.0 

52-3    59-6 

69.2 

24 

o 

4.2      8.4     12.6 

17.0 

21.4 

26.0  30.7  35.7 

41.0 

46.9 

534    61.1 

71.6 

26 

o       4.2  :    8.5 

12.8 

*7-3 

21.8 

26.4!  31.3  1  36.4 

41.9 

47-9 

54-7    62.9 

74.6 

28 

o  |    4.3      8.7 

I3-I 

17.6 

22.2 

26.9  31.9 

37-2 

42.8 

49.0 

56.1 

65.0 

79.0 

3O 

o.o       4.4      8.8 

r3-3 

17.9 

22.6 

27.5  32.6 

38.0 

43-8 

5°-3 

57-9    67.5 

31 

o       4.5      8.9 

13-5 

18.1 

22.9 

27-8i  33-o    38-5    444 

51.0 

58.8    68.9 

32 

o       4.5      9.0 

13-6 

18.3 

23.1 

28.1    3341  39-°    45-° 

51.8 

59-8    70.6 

33 

o 

4.6     9.1 

13.8 

18.5 

234 

28-5    33-8    39-5    45-7 

52.6 

61.0    72.5 

34 

o 

4.6     9.2 

14.0 

18.8 

23-7 

28.8  1  34.3    40.0    46.4 

53-5 

62.2     74.8  1 

| 

35 

0.0          4-7  !       9.4 

14.1 

19.0 

24.0 

29.2    34.7    40.6 

47.1 

54-5 

63-5 

77.6 

36 

0         4.7  !      9.5 

H-3 

19.2 

24-3 

29.6;  35.2    41.2 

47-9 

55-5 

65.0 

81.4  i 

37 

o       4.8  j     9.6]  14.5 

19-5 

24.7 

30.0  1  35.7    41.9    48.7 

56.6 

66.7 

|          j 

38 

•  o       4.8      9.7     14.7 

19.8 

25.0 

3°-5  i  36-3    42.6  i  49.6 

57-8 

68.5 

1 

39 

o       4.9  j     9.9 

14.9 

20.  i 

254 

31.0;  36.9;  43.3    50.5 

59-i 

70.7 

1  . 

4O 
41 

O.O 
0 

5-o 
5-i 

IO.O 
10.2 

i5-i 
J54 

20.4 
20.7 

25-8 
26.2 

3i-5  i  37-5 
32.0  !  38.2 

44.1 
45-o 

51.6 
52.6 

60.5 
62.0 

£3 

42 

o       5-2 

10.3 

15.6 

21.2 

26.6 

32.6    38.9     45.9 

53-8 

63.8 

80.7 

43 

o       5.2     10.5 

15-9 

21.4 

27.1 

33-  i 

39-6 

46.8 

55-i 

65-7 

44 

o 

5-3  I  io-7 

16.1 

21.8 

27.6 

33-8 

40.4 

47-8 

56.5 

67.9 

45 

o.o       5.4 

IO-9 

16.4 

22.1 

28.1 

344 

41.3    49:0 

.58.0 

70.5 

46 

o       5.5     ii.  i 

16.7 

22.6 

28.7 

35-1 

42.2 

50.2 

59-7 

73-7 

47 

o       5.6     11.3 

17.0 

23.0 

29-3 

35-9    43-2 

514 

77-8 

48 
49 

o       5-7 
o  i     5.8 

"•5 
1.1.7 

17.4 

17.7 

23-5 
24.0 

29.9 
30.5 

36.7 
37-6 

44.21  52.9 
45-3    544 

63^7 
66.1 

85.1 

50 

o.o 

6.0 

12.0 

18.1 

24-5 

31.2 

38.5 

46.5 

56.1 

69.0 

51 

o 

6.1 

12.2 

18.5 

25.1 

32.0 

39-5 

47-9 

57-9 

72.4 

52 

o 

6.2 

12.5 

18.9 

25-7 

32.8 

40-5 

49-3 

60.0 

77.0 

53 

o 

6.4 

12.8 

19.4 

26.3 

33-6 

41.6 

50.8 

62.4 

85.6 

54 

o 

6.5 

!3-i 

19.9 

27.0 

34-5 

42.9 

52.5 

65.1 

55 

0.0 

6.7 

134 

20.4 

277 

35-5 

44.2 

544 

68.4 

56 

o 

6.8 

13.8 

20.9 

28.5 

36.6 

45-7 

56.6 

72.5 

57 

o 

7-o 

14.2 

21.5 

'    29-3 

37-7 

47-3 

59-o 

78.3 

58 

0 

7.2 

14.6 

22.2 

30.2 

39-o 

49.0 

61.7 

59 

0 

74 

15.0 

22.8 

31.2 

40-3 

5°-9 

65.0 

60 

o.o 

7-7 

*5-5 

23.6 

32.2 

41.8 

53-i 

69.0 

61 

o 

7-9 

16.0 

24.4 

334 

434 

55.6 

74-3 

62 

0 

8.2 

16.5 

25.2 

!  34-6 

45-2 

584 

83-7 

63 

0 

8.4 

17.1 

26.1 

36.0 

47.2 

61.8 

64 

o 

8.7 

17.7 

27.1 

37-5 

49-5 

65.8 

65 

o.o 

9.1 

18.4 

28.2 

39-i 

52.1 

71.2 

66 

0 

9.4 

19-1 

29.4 

41.0 

55-° 

79-5 

67 

o 

9.8 

19.9 

30.8 

!  43-° 

68 

o 

10.3 

2O.8 

32-3 

'  454 

62^8 

69 

0 

10.7 

21.8 

33-9 

48.1 

68.5 

7O 

o.o 

II.  2 

22.9 

35-8 

!  51-2 

77.1 

71 

0 

1  1.8 

24.2 

37-9 

55-° 

72 

0 

12.5 

25.6 

40-3 

59-6 

73 

o 

13.2 

27.1 

43-2 

65.8 

74 

0 

14.0 

28.9 

46.5 

I 

I 

! 

| 

» 

The  Azimuths  of  this  Table  correspond  to  the  stated  changes  in  the  Altitude  of 
a  Heavenly  Body  in  One  Minute  of  Time. 

160 


TABLE  Lll. 


Direct  Bearings  of  a  Fixed  Object  :   Limiting  Distance  of  the  Object. 

1                                                                                                                                                            i 

Si 

2  a 

Distance-Ratio,  or  Distance  of  the  Object  divided  by  the  Radius  of  owing. 

Sj? 

!  If, 



ll  ' 

K® 

30 

40 

50 

75 

100 

150 

200 

250 

300 

.{.*!> 

400 

50O 

600 

700  SOU  90O  1OOO 

K® 

Ftet. 

\.M 

.V.  -V. 

.v.  jr. 

V    V 

.V.  X 

A'.  .V. 

A.  M. 

.V.  M. 

A    .!/.       A".  M. 

A'.  M. 

A.  .»/. 

A    .)/      V.  M.     \.    \J     .V.  M.        \.    M. 

,,.,.    1 

50 

0.2 

0-3 

0.4 

0.6 

0.8 

1.2 

1.6 

2.1 

2.S 

2.9 

3-3 

4 

5       6       7       7  I      8 

50 

75 

0.4 

0.6 

0.9 

1.2 

1.8 

2-S 

3-i 

3-7 

4-3 

4.9 

6 

9  i  10      ii 

12 

75 

1  1OO 

O-5 

0.7 

0.8 

1.2 

1.6 

2.S 

3-3 

4.1 

4.9 

S-7 

6.6 

8 

10 

ii 

IT 

JC 

16 

100 

125 

0.6 

0.8 

I.O 

1.5 

2.0 

4.1 

S-i 

6.1 

7.2 

8.2 

10 

12 

14 

16 

18 

20 

125 

150 

0.7 

I.O 

.2 

1.8 

2-5 

3-7 

4.9 

6.2 

7-4 

8.6 

9.9 

12 

'5 

17 

2O       22 

25 

15O 

175 

0.9 

1.2 

•4 

2.2 

2.9 

4-1 

S-7 

7.2 

8.6 

IO.I 

ii.  5 

14 

17 

20 

23     26 

29 

175 

200 

I.O 

1.3 

.6 

2.4 

3-3 

4-9 

6.6 

8.2 

9-9 

II-5 

13.1 

16 

2O 

23 

2b      30 

33 

2OO 

'  225 

i.i 

'•5 

.8 

2.8 

3-7 

S-S 

7-4 

9.2 

ii.  I 

12.9 

14.7 

18 

22 

26 

30 

33 

225 

250 

1.2 

1.6 

2.O 

3-1 

4.1 

6.2 

8.2 

10.3 

12.3 

14.4 

16.4 

21 

25 

29 

33 

38 

42 

250 

275 

1.4 

1.8 

2-3 

3-4 

4-5 

6.8 

9.0 

1  1-3!  13-5 

15.8 

1  8.0 

23 

27 

32 

36 

45 

275 

300 

*•$ 

2.0 

2.S 

3-7 

4.9 

7-4 

9.9 

12.3  !  14-8 

17.2 

19.7 

2S 

30 

34 

39 

44 

49 

30O 

325 

1.6 

2.1 

2.7 

4.0 

.5-3 

8.0 

10.7    13.3  1  16.0 

18.7 

21.4 

27 

32 

37 

43     48 

325 

350 
375 
400 

i!8 

2.0 

2-3 

2.9 

3-3 

4-3 
4.6 
4.9 

5.8 

6.2 

6.6 

8.6 
9.2 
9-9 

11.5 
12.3 

14.4 

'5-4 
16.4 

17.2 
18.5 
19.7 

20.  i 

21.6 

23.0 

23.0 
24.6 
26.3 

29 
31 

33 

34 
37 
40 

40 
43 

47 

46 
49 
53 

52 

1 

67 

350 
375 
4OO 

The  Distances  are  expressed  in  Nautical  Miles  of  6,086  feet. 

1 

TABLE  LIII. 

Direct  Bearings  of  a  Fixed  Object:  Parallactic  Errors  of  the  Bearings. 

3.S 

Distance-Ratio,  or  Distance  of  the  Object  divided  by  the  Radius  of  Swing. 

*«, 

o  A 

|l 

30 

40 

50 

75 

100 

150 

200 

250 

300 

350 

400 

500 

600 

7OO 

800  900 

1000 

Is 

o 

0 

o 

o 

0 

0 

0 

0 

0 

0 

o 

0 

o 

o 

0 

0 

'0 

0 

0      | 

0 

O.O 

O.O 

O.O 

O.O 

O.O 

O.O 

0.0 

O.O 

0.0 

0.0 

O.O 

O.O 

O.O 

0.0 

O.O 

O.O 

O.O 

O; 

10 

3 

2 

2 

I 

I 

I 

I 

0 

0 

0 

0 

0 

O 

0 

o 

O 

o 

10 

2O 

7 

S 

4 

3 

2 

I 

I 

I 

I 

O 

o 

o 

O 

o 

0 

0 

0 

20  i 

30 

I.O 

7 

6 

4 

3 

2 

I 

I 

I 

I 

I 

O 

o 

0 

O 

0 

30  i 

40 

2 

9 

7 

5 

4 

2 

2 

I 

I 

I 

I 

I 

I 

o 

o 

o 

1O 

50 

i-S 

i.i 

0.9 

0.6 

0.4 

0-3 

0.2 

0.2 

O.I 

O.I 

0. 

O.I 

o. 

o. 

O.I 

O.I 

0.0 

50 

6O 

7 

2 

I.O 

7 

s 

3 

2 

2 

2 

I 

I 

I 

o 

oo 

70 

8 

3 

i 

7 

4 

3 

2 

2 

2 

I 

I 

I 

70 

§0 

9 

4 

2 

8 

6 

4 

3 

2 

2 

2 

I 

I 

I 

80  ! 

90 

1.9 

1.4 

1.2 

0.8 

0.6 

0.4 

0.3 

0.2 

O.2 

0.2 

O.2 

O.I 

0. 

0. 

O.I 

0. 

O.I 

90 

100 

1.9 

1.4 

1.2 

0.8 

0.6 

0.4 

0.3 

O.2 

O.2 

0.2 

O.I 

O.I 

o. 

o. 

0.1 

0.. 

O.I 

100 

110 

8 

3 

I 

7 

s 

4 

3 

2 

2 

2 

I 

I 

I 

I 

110 

120 

7 

2 

0 

7 

5 

3 

2 

2 

2 

I 

I 

I 

I 

0 

120 

13O 

5 

I 

0.9 

6 

4 

3 

2 

2 

I 

I 

'    I 

I 

0 

130 

i 

140 

1.2 

O.q 

0-7 

0.5 

0.4 

0.2 

0.2 

O.I 

O.I 

O.I 

O.I 

O.I 

O.I 

O.I 

O.Q. 

0.0 

0-0 

140 

15O 

0 

7 

6 

4 

3 

2 

I 

I 

I 

I 

I 

I 

o 

o 

o 

o 

0 

150 

I  16O 

0.7 

S 

4 

2 

I 

I 

I 

I 

O 

o 

0 

0 

0 

o 

o 

o 

1  <»O 

170 

3      2 

j 

1 

I 

I 

0 

O 

0 

o 

o 

0 

0 

0 

o 

o 

170 

!  180 

o 

o 

0 

0 

O 

o 

0 

0 

0 

0 

0 

o 

o 

0 

0 

0 

o 

ISO; 

.      -j 
The  Parallactic  Error  is  applied  to  the  Right  or  Left  of  the  Compass-Error  (Total  ! 

or  Deviation)  according  as  the  Angle  of  Swing  falls  to  the  Left  or  Right  of  the  • 

Bearing  of  the  Object  ;   that  is,  contrariwise  to  the  Angle  of  Swing. 

TABLE  LIV. 


1G1 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Sines  of  the  Rhumbs. 

Arcs. 

So  =    1    Si  = 

S,  =        83  = 

S4  = 

85  = 

S6  = 

S7  = 

S8  = 

Arcs.  | 

0.000  0.195 

0.383    0.556 

O.7O7 

O.831    O.924 

O.981 

l.OOO 

o 

o                 o 

0                           0 

o 

o 

o 

o 

o 

o 

o.o 

o.o    '     o.oo         o.oo          o.oo 

o.oo 

0.00 

O.OO 

0.00 

o.oo 

o.o 

1 

o 

02            04     i        06 

07 

08             09 

10 

IO 

1 

2 

o 

04           08    j        II 

H 

17             18             20             20 

3 

o 

06           II            17 

21 

25             28             29             30 

O      i 

4 

0 

08 

I5      j           22 

28 

33            37 

39            40 

4 

O.5 

o.o 

O.IO 

0.19 

0.28 

o-35 

0.42 

0.46 

0.49 

'0.50 

0.5 

6 

o 

12 

23 

33 

42 

50 

55 

59 

60 

6 

7 

o 

H 

27    !        39 

49 

58 

65 

69 

70 

7 

8 

o 

16 

31 

44 

57 

66 

74 

78 

80 

8 

9 

o    |        ,8 

34 

5° 

64 

75 

83 

88 

90 

9 

1.0 

o.o 

0.20 

0.38          0.56 

0.71 

0.83 

092 

0.98 

I.OO 

1.0 

1 

o 

21 

42 

61 

78 

91 

1.02 

i.  08 

IO 

1 

2 

o 

23 

46 

67 

85 

I.OO 

II 

18 

20 

SJ 

3 

o 

25 

5°            72 

92 

08 

20 

28 

30 

3 

4 

o 

27 

54            78 

99 

16 

29 

37 

40 

4 

1.5 

6 

o.o 
o 

0.29 
31 

0.57         0.83 
61             89 

i.  06 
13 

1.25 
33 

1-39 
48 

1.47 

12 

1.5 

6 

7 

o 

33 

65             94 

20 

4i            57 

67 

70 

7 

8 
9 

0 

o 

35 
37 

69          i.oo 
73            06 

27 

34 

5°            66 

58            75 

U 

80 
90 

9 

2.0 

0.0 

0.39   I    0.77       1.  1  1 

1.41 

1.66          1.85 

1.96 

2.00 

2.0 

1 

o 

41 

80             17 

48 

75            94 

2.06 

IO 

1 

2 

o 

43 

84                 22 

56 

83          2.03 

16 

20 

2 

3 

o 

45 

88 

28 

63 

91                  12 

26 

30 

3 

4 

o 

47 

92 

33 

70 

99            22    |        35 

40 

4 

2.5 

0.0 

0.49 

0.96 

1-39 

1.77 

2.07          2.31     !      2.45 

2.50 

2.5 

6 

o 

51 

i.oo            44 

84 

16            40    !        55 

60 

6  i 

7 

o 

53 

03             50 

24            49 

65 

70 

7 

*  8 

0 

55 

07            56 

98 

33            59 

75 

80 

8 

9 

o 

57 

ii             61 

2.05 

41             68 

84 

90 

9 

3O 

o.o 

°-59 

1.15           1.67 

2.12 

2.49          2.77 

2.94 

3.00 

3.0 

1 

0 

60 

19 

72 

*9 

58            86 

3-°4 

IO 

1  I 

2 

o 

62 

22 

78 

26 

66 

95 

20 

2  ! 

3 

o 

64 

26 

83 

33 

74 

3-05 

24 

30 

3 

4 

o 

.  66 

30 

89 

40 

83 

14 

33 

40 

4 

3.5 

o.o 

0.68 

i-34 

J-95 

2.47 

2.91 

3-23 

343 

3.50 

3.5 

6 
7 

o 

0 

70 
72 

38 

42 

2.00 

06 

II 

33 

42 

g 

z° 

6 

7 

8 

o 

74 

45 

II 

69 

3'i6 

73 

So 

8 

9 

o 

76 

49 

17 

76 

24 

60 

83 

90 

9 

4.0 

0.0 

0.78 

1.53 

2.22 

2.83 

3-33 

3-69' 

3-92 

4.00 

4.0 

I 

o 

80 

57 

28 

90 

4.02 

IO 

1 

2 

o 

82 

61 

33 

97 

49 

88 

12 

20 

2 

4 

o 

0 

84 

86 

65 
68 

39 
44 

3-04 
ii 

8. 

97 

4.06 

22 
32 

3° 

40 

3 
4 

4.5 

o.o 

0.88 

1.72 

2.50 

3.18 

3-74 

4.16 

4.41 

4.50 

4.5 

6 

o 

90 

76 

56 

25 

82 

25 

51 

60 

6 

7 

o 

92 

79 

61 

32 

91 

34 

61 

70 

7 

8 

o     |        94 

83 

67 

39 

99 

43 

71 

80 

8  ! 

9 

o    j        96 

87 

72 

46 

4.07 

53 

81 

90 

9 

5.0 

o.o    i     0.98 

1.91 

2.78 

3-54 

4.16 

4.62 

4.90 

5.00 

5.0 

1 

o 

I.OO 

95 

83 

61 

24 

7i 

5.00 

10 

1 

2 

o 

OI 

99 

89 

68 

32 

80 

10 

20 

2 

3 

o 

03 

2.03 

95 

75 

40 

90 

20 

30 

3 

4 

o 

°5 

07 

3.00 

82 

49 

99 

30 

40 

4 

5.5 

o.o 

1.07 

2.10 

3.06 

3-89 

4-57 

5.08 

5-39 

5-50 

5.5 

6 

o 

09 

J4 

ii 

96 

66 

17 

49 

60 

6 

7 

o 

ji 

18 

17 

4-03 

74 

27 

59 

70 

7 

§ 

o 

13 

22 

22 

10 

82 

36 

69 

80 

8 

9 

0 

*5 

26 

28 

17 

9i 

45 

79 

90 

9 

6.0 

0.0 

1.17 

2.30 

3-33 

4.24 

4-99 

5-54 

5.89 

6.00 

6.0 

162 


TABLE  L1V. 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

So- 

o.ooo 

s,  = 

0.195 

0.383 

O.556 

0.707 

0.831  j  0.924 

0.981 

l.OOO 

6°0 

o 

o.o 

0 

1.17 

0 

2.30 

o 

3-33 

o 

4.24 

o                   o 

4-99          5-54 

o 

o 

6.00 

o 

6.0 

1 

0 

19 

33 

39 

31 

5-°7            63 

98 

10 

1 

2 

o 

21 

37 

44 

38 

15    \        73 

6.08 

20 

2 

3 

0 

23 

50 

46 

24            82 

18    i       30 

3 

4 

0 

25 

45 

55 

53 

32    j        91            28            40 

4 

6.5 

0.0- 

1.27 

2-49 

3.61 

4.60 

5.40         6.00    !     6.37    ;     6.50 

6.5 

6 

o 

29 

52 

67 

67 

48            09    !        47    ;        60 

6 

7 

o 

31 

56 

72 

74 

57            19            57    i        7o 

7 

0 

33 

60 

78 

81 

65 

28         67   ;      80 

8 

9 

o 

34 

64 

83 

88 

74 

37            77 

90 

9 

7.0 

o.o 

1.36 

2.68 

3-89 

4-95 

5.82 

6.47         6.87 

7.00 

7.0 

1 

o 

38 

72 

94 

5-02 

90 

56            96 

10 

1 

2 

0 

40 

75 

4.00 

09 

99 

65 

7.06 

20 

2 

3 

o 

42 

79 

05 

16 

6.07 

74 

16 

30 

3 

4 

o 

44 

83 

ii 

23 

15 

84 

26 

40 

4 

7.5 

o.o 

1.46 

2.87 

4.17 

5-30 

6,24 

6-93 

7.36 

7.50 

7.5 

6 

o 

48 

22 

37 

32 

7.02 

45 

60 

6 

7 

o 

5° 

95 

28 

44 

40 

ii 

55 

70 

7 

o 

52 

98 

33 

51 

48 

21 

65 

80 

8 

9 

0 

54 

3.02 

39 

58 

57 

30 

75 

90 

9 

8.0 

0.0 

1.56 

3-06 

4-44 

5-65 

6.65 

7-39 

7.85 

8.00 

8.0 

1 

0 

58 

10 

50 

73 

73 

48 

94 

10 

1 

2 
3 

0 
0 

60 
62 

[J 

1! 

80 
87 

82 
90 

f 
67 

8.04 

20 

30 

2 
3 

4 

o 

64 

21 

67 

94 

98 

76 

24 

40 

4 

8.5 

o.o 

1.66 

3-25 

4.72 

6.01 

7.07 

7.85 

8-34 

8.50 

8.5 

6 

0 

68 

29 

78 

08 

15 

94 

43 

60 

6 

7 

0 

70 

33 

83 

15 

23 

8.04 

53 

70 

7 

8 

o 

72 

37 

89 

22 

32 

13 

63 

80 

.  8 

9 

0 

74 

95 

29 

40 

22 

73 

90 

9 

9.0 

0.0 

1-75 

3-44 

5-oo 

6.36 

7-48 

8.3I 

8.83 

9.00 

9.0 

1 

o 

77 

48 

05 

43 

56 

41 

92 

10 

1 

,    2 

o 

79 

52 

ii 

51 

65 

50 

9.02 

20 

2 

3 
4 

0 
0 

81 
83 

f 
60 

17 

22 

58 
65 

8 

12 

22 

30 
40 

3 
4 

9.5 

o.o 

1.85 

3-63 

5.28 

6.72 

7.90 

8.78 

9-32 

9.50 

9.5 

6 

o 

87 

67 

33 

79 

98 

87 

42 

60 

6 

7 

o 

39 

86 

8.06 

96 

51 

70 

7 

8 

o 

I 

75 

44 

93 

15 

9-05 

61 

80 

8 

9 

o 

93 

79 

50 

7.00 

23 

15 

71 

90 

9 

10.0 

o.o 

1-95 

3-83 

5-56 

7.07 

8.31 

9.24 

9.81 

10.00 

10.0 

1 

.0 

97 

86 

61 

14 

40 

33 

91 

IO 

1 

2 

o 

99 

90 

67 

21 

48 

42 

10.00 

20 

2 

3 

o 

2.01 

72 

28 

56 

5i 

IO 

30 

3 

4 

0 

03 

98 

78 

35 

65 

61 

20 

40 

4 

1O.5 

0.0 

2.05 

4.01 

5-83 

7.42 

'    8.73 

9.70 

10.30 

10.50 

10.5 

6 

0 

07 

05 

89 

50 

81 

79 

40 

60 

6 

7 

o 

09 

09 

94 

57 

90 

oo 

49 

70 

7 

o 

ii 

13 

6.00 

64 

98 

•    98 

59 

80 

8 

9 

0 

13 

17 

05 

7i 

9.06 

10.07 

69 

90 

9 

11.0 

0.0 

2.15 

4.21 

6.ii 

7.78 

9-15 

10.16 

10.79 

11.00 

11.0 

1 

o 

16 

25 

17 

85 

23 

25 

89 

10 

1 

2 

o 

18 

29 

22 

92 

35 

98 

20 

2 

3 

o 

20 

32 

27 

99 

39 

44 

1  1.  08 

3° 

3 

4 

0 

22 

36 

33 

8.06 

48 

53 

18 

40 

4 

11.5 

o.o 

2.24 

4.40 

6-39 

8.13 

9.56 

10.62 

11.28 

11.50 

11.5 

6 

0 

26 

44 

44 

20 

64 

72 

38 

60 

6 

7 

o 

28 

48 

5° 

27 

73 

81 

47 

7o 

7 

8 

0 

3° 

51 

56 

34 

81 

90 

57 

80 

8 

9 

o 

32 

55 

61 

41 

89 

99 

67 

90 

9 

12.0 

0.0 

2-34 

4-59 

6.67 

8-49 

9.98 

11.09 

11.77 

12.00 

190 

TABLE  LIV. 


163 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

o.ooo 

Si  — 
0.195 

O.383 

«       

0.556 

84  = 

0.707 

0.831 

O.924 

0.981 

1.000 

Q 

o 

o 

o 

o 

o 

o 

o 

•  o 

o 

120 

o.o 

2.34 

4-59 

6.67 

8.49 

9.98 

11.09 

11.77 

12.00 

12.0 

1 

o 

36 

63 

72 

56 

1  0.06 

18 

87 

10 

1 

2 

0 

•    38 

67 

78 

63 

H 

27 

97 

20 

2 

3 

o 

40 

83 

70 

23 

36 

12.06 

3° 

3 

4 

o 

42 

74 

89 

77 

46 

16 

40 

4 

125 

0.0 

2.44 

4-78 

6.94 

8.84 

10.39 

"•55 

12.26 

12.50 

12.5 

6 

0 

46 

82 

7.00 

91 

48 

64 

36 

60 

6 

7 

0 

48 

86 

05 

'   98 

56 

73 

46 

70 

7 

8 

o 

50 

90 

II 

9-05 

64 

82 

56 

80 

8 

9 

0 

52 

94 

16 

12 

72 

92 

65 

90 

9 

13.0 

0.0 

2.54 

4.98 

7.22 

9.19 

10.81 

12.01 

12-75 

13.00 

13.0 

1 

0 

56 

5.01 

28 

26 

89 

10 

85 

10 

1 

2 

o 

57 

05 

33 

33 

97 

19 

95 

20 

2 

3 

o 

59 

09 

39 

40 

1  1.  06 

29 

13-05 

30 

3 

4 

o 

61 

12 

44 

47 

J4 

38 

15 

40 

4 

13.5 

o.o 

2.63 

5.l6 

7-50 

9-54 

11.23 

12.47 

13.24 

13.50 

13.5 

6 
7 

0 

o 

65 
67 

20 
24 

II 

61 

69 

39 

£ 

34 
44 

60 
70 

6 

7 

0 

69 

28 

67 

76 

47 

75 

54 

80 

8 

9 

0 

32 

72 

83 

56 

84 

63 

90 

9 

140 

o.o 

2-73 

5.36 

7.78 

9-90 

11.64 

12.93 

13-73 

14.00 

14.0 

1 

0 

75 

39 

83 

97 

72 

i3-03 

83 

10 

1 

2 

0 

77 

43 

89 

10.04 

9i 

12 

93 

2O 

2 

3 

o 

79 

47 

94 

ii 

89 

21 

14.03 

30 

3 

4 

0 

81 

51 

8.00 

18 

97 

30 

13 

40 

4 

14.5 

0.0 

2.83 

5-55 

8.06 

10.25 

12.06 

1340 

14.22 

14.50 

14.5 

6 

7 
8 
9 

o 

0 
0 
0 

85 
87 
89 

59 
63 
67 

70 

ii 

17 

22 
28 

32 
39 
46 

54 

14 

22 
31 

39 

49 
f 
67 

77 

32 

42 

g 

60 
70 
80 
90 

6 
7 

8 
9 

15.0 

0.0 

2-93 

5-74 

8-33 

10.61 

12.47 

13.86 

14.71 

I5.OO 

15.0 

1 

o 

78 

39 

68 

56 

95 

81 

10 

1 

2 

o 

96 

82 

44 

75 

64 

14.04 

91 

20 

2 

3 
4 

0 

o 

3.00 

85 
89 

50 
55 

82 
89 

B 

13 

23 

15.01 

10 

30 
40 

3 
4 

15.5 

o.o 

3-02 

5-9? 

8.61 

10.96 

12.89 

14.32 

15.20 

15.50 

15.5 

6 

0 

04 

97 

67 

11.03 

97 

3° 

60 

6 

7 

o 

06 

6.01 

72 

10 

5° 

40 

70 

7 

8 

o 

08 

°5 

78 

*7 

H 

60 

CQ 

80 

8 

9 

0 

10 

09 

83 

24 

22 

69 

60 

90 

9 

16.O 

o.o 

3.12 

6.12 

8.89 

11.31 

13.30 

14.78 

15-69 

16.00 

16.O 

1 

o 

H 

16 

94 

38 

38 

87 

79 

10 

1 

2 

0 

16 

20 

9.00 

45 

47 

97 

89 

20 

2 

3 
4 

o 

0 

18 

20 

24 
27 

06 
ii 

g 

15.06 
15 

1  6.08 

30 
40 

3 
4 

16.5 

0.0 

3.22 

6.31 

9.17 

11.67 

13.72 

15.24 

16.18 

16.50 

16.5 

6 

0 

24 

35 

22 

74 

80 

34 

28 

DO 

6 

7 

o 

26 

39 

28 

81 

•    88 

43 

38 

70 

7 

8 

o 

28 

43 

33 

88 

97 

52 

80 

8 

9 

o 

30 

47 

39 

95 

14.05 

61 

58 

90 

9 

17.0 

0.0 

3-32 

6.51 

9-44 

12.02 

14.14 

'15.71 

16.68 

17.00 

17.0 

1 

0 

34 

54 

5° 

09 

22 

80 

77 

10 

1 

2 

0 

35 

58 

55 

16 

3° 

89 

87 

20 

2 

3 

o 

37 

62 

61 

23 

38 

98 

97 

30 

3 

4 

0 

39 

66 

67 

30 

47 

16.08 

17.07 

40 

4 

17.5 

o.o 

3-41 

6.70 

9.72 

12.37 

14-55 

16.17 

17.16 

17.50 

17.5 

6 

0 

43 

73 

78 

45 

63 

26 

26 

60 

6 

7 
8 
9 

o 

0 
0 

45 
47 
49 

11 
85 

83 
89 
94 

52 

31 

g 

88 

35 
44 
54 

36 
46 

56 

70 
80 
90 

8 
9 

18.0 

0.0 

3-51 

6.89 

10.00 

12.73 

14.97 

16.63 

17-65 

18.00 

18.0 

164 


TABLE  LIV. 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

:  Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

o.ooo 

s,= 

0.195 

0.383 

0.556 

0.701 

0.831 

So  = 
0.924 

0.981 

l.OOO 

o 

o 

o 

0 

o 

o 

0 

o 

0 

o 

o 

1   18.0 

o.o 

3.5  I 

6.89 

IO.OO 

12.73 

14.97 

16.63 

18.00 

18.O 

1 

o 
o 

53 

55 

Si 

06 

II 

80 

87 

'3 

e 

11 

IO 

20 

1   i 
2 

3 

0 

57 

7.00 

17 

94 

22 

91 

3° 

3 

4 

0 

59 

04 

22 

13.01 

30 

17.00 

18.05 

40 

4 

18.5 

o.o 

3.61 

7.08 

10.28 

13.08 

I5-38 

17.09 

18.14 

18.50 

18.5  : 

6 

o 

63 

12 

33 

15 

46 

18 

24 

60 

6 

7 

0 

65 

15 

39 

22 

55 

27 

34 

70 

7 

g 

o 

67 

19 

44 

29 

63 

37 

44 

80 

8 

9 

o 

69 

23 

50 

36 

46 

54 

90 

9 

19.0 

o.o 

3.71 

7.27 

10.56 

13-43 

15.80 

17-55 

18.64 

19.00 

19.O 

1 

o 

73 

31 

61 

5° 

88 

64 

73 

IO 

• 

2 

o 

74 

35 

67 

58 

96 

74 

83 

20 

2 

3 

o 

76 

39 

72 

65 

16.05 

83 

93 

30 

3 

4 

0 

78 

42 

78 

72 

13 

92 

19.03 

40 

4 

19.5 

0.0 

3.80 

7.46 

10.83 

13.79 

16.21     i    18.01 

I9-X3 

19.5° 

19.5   \ 

6 

o 

82 

50 

89 

86 

30 

ii 

22 

60 

6 

7 

0 

84 

94 

93 

38 

20 

32 

70 

* 

8 

0 

86 

58 

11.00 

14.00 

46 

29 

42 

80 

8  I 

9 

o 

88 

62 

06 

07 

55 

39 

52 

90 

9 

20.0 

o.o 

3-90 

7.66 

ii.  ii 

14.14 

16.63 

18.48 

19.62 

20.00 

20.0 

1 

o 

92 

69 

17 

21 

7i 

57 

72 

IO 

1   1 

2 

o 

94 

73 

22 

29 

80 

66 

81 

20 

2  j 

3 

o 

96 

77 

28 

36 

88 

75 

9i 

30 

3 

4 

0 

98 

81 

33 

43 

96 

84 

20.01 

40 

4   ! 

i  20.5 

0.0 

4.00 

7.84 

11-39 

14.50 

17.05 

i8.94 

20.11 

20.50 

20.5 

6 

o 

02 

88 

45 

57 

13 

19.03 

21 

60 

6 

7 

8 

0 

o 

3 

92 
96 

56 

64 
71 

21 
29 

12 
22 

30 
40 

1° 

7 

8 

9 

0 

08 

8.00 

61 

78 

38 

31 

50 

90 

9  j 

21.0 

0.0 

4.10 

8.04 

11.66 

14.85 

17.46 

I9-40 

20.59 

21.00 

21.0 

1 

0 

12 

08 

72> 

92 

54 

49 

69 

IO 

1 

2 
3 

o 
o 

13 
15 

ii 
15 

78 
83 

99 
15.06 

62 

11 

89 

20 

3° 

2 
3 

4 

0 

17 

19 

89 

13 

79 

77 

99 

40 

4 

21.5 

0.0 

4.19 

8.23 

11.94 

15.20 

17.87 

,9.86 

21.09 

21.50 

21.5 

6 

o 

21 

27 

12.00 

28 

96 

96 

19 

60 

6 

7 

0 

23 

30 

06 

35 

18.04 

20.05 

29 

70 

7 

8 

0 

25 

34 

II 

42 

13 

H 

38 

80 

8 

9 

o 

27 

38 

17 

49 

21 

23 

48 

90 

9 

22.O 

1 

o.o 

0 

4.29 
31 

8.42 
46 

12.22 
28 

I5-56 
63 

l8.29 

37 

20.33 
42 

"if 

22.00 
IO 

22.O 
1 

2 

o 

33 

50 

33 

70 

46 

5i 

78 

20 

2 

3 

o 

35 

53 

39 

77 

54 

60 

87 

3° 

3 

4 

0 

37 

57 

45 

84 

62 

69 

97 

40 

4 

22.5 

0.0 

4-39 

8.61 

12.50 

I5-9I 

18.71 

20.79 

22.07 

22.50 

22.5 

6 

o 

41 

65 

56 

98 

79 

88 

17 

60 

6 

7 

o 

43 

69 

61 

16.05 

Q 

97 

26 

70 

7 

8 

0 

45 

72 

67 

12 

96 

21.06 

36 

80 

8  ! 

9 

0 

47 

76 

72 

19 

19.04 

15 

46 

90 

9 

23.0 

0.0 

4-49 

8.80 

12.78 

16.26 

19.12 

21.25 

22-56 

23.00 

23.0 

1 

o 

51 

84 

83 

34 

21 

34 

66 

10 

1 

2 

0 

53 

88 

89 

29 

43 

76 

20 

2 

3 
4 

0 
0 

II 

92 
95 

95 

13.00 

48 
55 

46 

e 

95 

3° 

40 

3 
4 

23.5 

0.0 

4-58 

8-99 

13.06 

16.62 

19.54 

21.71 

23-05 

23.50 

23.5 

6 

o 

00 

9-03 

ii 

69 

62 

80 

15 

60 

6 

7 

o 

62 

07 

17 

76 

71 

90 

25 

70 

7 

8 

o 

64 

ii 

22 

83 

79 

99 

35 

80 

8 

9 

0 

66 

15 

28 

90 

°7 

22.08 

45 

90 

9 

24.0 

0.0 

4.68 

9.18 

'3-33 

16.97 

19.95       22.17 

23-54 

24.00 

24.0 

TABLE  LIV. 


165 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

£t         £i        £«        £•        

0.000  0.195  J0.383    O.556 

0.707 

0.831    0.924 

g   
O.981 

l.OOO 

240 

o                   o 

o.o         4.68 

0                            0 

9-18     13.33 

o 

16.97 

o                    o 

19.95       22.17 

23°54 

o 

24.00 

24.0 

1 

o             70 

22            39 

17.04 

20.03           26 

64 

IO 

1 

2 

o             72             26            44 

II 

12                 36 

74 

20 

2 

3 

o             74             30             50 

18 

20            45 

83 

3° 

3 

4 

o 

76 

34 

56 

25 

29 

54 

93 

40 

4 

24.5 

o.o 

4.78 

9.38 

13.61 

17.32 

20.37 

22.63 

24-03 

24.50 

24.5 

6 

o 

80            41             66 

39 

45 

72 

'3 

60 

6 

7 

o             82             45             72 

46 

53 

82 

23 

70 

7 

8 

o             84            49             78 

54 

62 

91 

33 

80 

8 

9 

o            86            53 

83 

61 

70 

23.00 

42 

90 

9 

25.0 

o.o         4.88 

9-57 

13.89 

17.68 

20.78 

23.09 

24.52 

25.00 

25.0 

1 

o 

90 

60 

94 

75 

87 

19 

62 

10 

1 

2 

o 

92 

64 

14.00 

82 

95 

28 

72 

20 

2 

3 

0 

93 

68 

05 

89 

21-03 

37 

81 

3° 

3 

4 

i 

o 

95 

72 

ii 

96 

ii 

46 

9i 

40 

4 

25.5 

o.o 

4-97 

9.76 

14.16 

18.03 

21.  2O 

23.56 

25.01 

25-50 

25.5 

6 

o 

99 

79 

22 

IO 

28 

65 

ii 

60 

6 

7 

o 

5.01 

83 

28 

17 

37 

74 

21 

70 

7 

8 

o 

°3 

87 

33 

25 

45 

83 

30 

80 

8 

9 

0 

°5 

91 

39 

32 

53 

93 

40 

90 

9 

26.0 

0.0 

5.07 

9-95 

14.44 

18.39 

21.62 

24.02 

25.50 

26.00 

26.O 

1 

o 

09 

99 

5° 

46 

70 

ii 

60 

10 

1 

2 

o 

ii 

53 

78 

20 

7° 

20 

2 

3 

0 

I3 

06 

61 

60 

87 

3° 

80 

3° 

3 

4 

o 

15 

IO 

67 

67 

95 

39 

89 

40 

4 

265 

o.o 

5.17 

10.14 

14.72 

18.74 

22.03    j   24.48 

25-99 

26.50 

26.5 

6 

0 

19 

18 

78 

81 

12 

57 

26.09 

60 

6 

7 

o 

21 

22 

83 

88 

20 

67 

19 

70 

7 

8 

0 

23 

25 

89 

95 

28 

76 

29 

80 

8 

9 

0 

25 

29 

94 

19.02 

36 

85 

38 

90 

9 

27.0 

o.o 

5.27 

IO-33 

15.00 

19.09 

22.45 

24-95 

26.48 

27.00 

27.O 

1 

o 

29 

37 

06 

16 

53 

25-04 

58 

IO 

1 

2 

o 

31 

ii 

23 

62 

13 

68 

20 

2 

3 

0 

33 

45 

17 

31 

7° 

22 

78 

3° 

3 

4 

o 

35 

48 

22 

38 

78 

31 

88 

40 

4 

27.5 

o.o 

5.36 

10.52 

15.28 

19-45 

22.86 

2540 

26.97 

27.50 

27.5 

6 

0 

38 

56 

33 

52 

95 

50 

27.07 

60 

6 

7 

o 

40            60 

39 

59 

23-03 

59 

17 

70 

7 

8 

o 

42    |        64 

44 

66 

ir 

68 

27 

80 

8 

9 

o            44    j        68 

50 

73 

20 

77 

37 

90 

9 

28.0 

o.o    i     5.46 

10.72 

15-55 

19.80 

23.28 

25-87 

27.47 

28.00 

28.0 

1 

2 

o 
o 

48 

75 
79 

61 
67 

87 
94 

36 

45 

96 
26.05 

II 

10 

20 

I 

2 

3 

o 

52 

83 

72 

2O.OI 

53 

15 

76 

3° 

3 

4 

o 

54 

87 

78 

08 

61 

.24 

86 

40 

4 

28.5 

o.o 

5'56> 

10.91 

15-83 

20.15 

23.69 

26.33 

27.95 

28.50 

28.5 

6 

0 

58 

94 

89 

22 

78 

42 

28.05 

60 

6 

7 

0 

60 

98 

94 

29 

86 

52 

15 

70 

7 

8 

o 

62 

11.02 

16.00 

37 

94 

61 

25 

80 

8 

9 

o 

64 

06 

°5 

44 

24.03 

70 

35 

90 

9 

29.O 

o.o 

5-66 

II.  10 

i6.n 

20.51 

24.11 

26.79 

28.44 

29.00 

29.0 

1 

o 

68 

13 

17 

58 

19 

88 

54 

IO 

1 

2 

o 

70 

17 

22 

65 

28 

98 

64 

20 

2 

3 

o 

72 

21 

28 

72 

36 

27.07 

74 

3° 

3 

4 

o 

74 

25 

33 

79 

44 

16 

84 

40 

4 

29.5 

o.o 

5-75 

11.29 

16,39 

20.86 

24.52 

27.25 

28.93 

29.50 

29.5 

6 

0 

77 

33 

44 

93 

61 

35 

29.03 

60 

6  1 

7 

o 

79 

36 

50 

21.00 

69 

44 

70 

7 

8 
9 

o 
o 

81 

83 

40 
44 

If 

07 

11 

23 
33 

80 
90 

8 
9 

30  0 

o.o 

5.85 

11.48 

1  6.66 

21.22 

24-94 

27.72 

29-43 

30.00 

30.0 

166 


TABLE  LIV. 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

o.ooo 

Si  = 
0.195 

83  = 
0.383 

S3  = 
0.556 

0.707 

S5  = 
0.831 

0924 

S7-_ 
0.981 

1.000 

30.0 

0 

o.o 

5°85 

0 

11.48 

i6°66 

0 

21.22 

24°94 

o 

27.72 

29-43 

30.00 

30.0 

1 

0 

'87 

52 

72 

29 

25-03 

Si 

53 

10 

1 

2 

o 

89 

56 

78 

36 

ii 

90 

62 

20 

2 

3 
4 

o 

0 

91 

93 

59 
63 

83 
89 

43 
50 

19 

28 

99 

28.09 

g 

30 
40 

3 
4 

30.5 

o.o 

5-95 

11.67 

26.94 

21-57 

25.36 

28.18 

29.92 

30-50 

30.5 

6 

0 

97 

71 

17.00 

64 

44 

27 

30.01 

60 

6 

7 

o 

99 

75 

05 

7i 

52 

36 

ii 

70 

7 

8 

0 

6.01 

79 

ii 

78 

61 

46 

21 

80 

8 

9 

o 

03 

83 

17 

85 

69 

55 

31 

90 

9 

31.0 

o.o 

6.05 

11.86 

17.22 

21.92 

25.78 

28.64 

30.41 

31.00 

31.O 

1 

o 

07 

90 

28 

99 

86 

73 

51 

10 

1 

2 

o 

09 

94 

33 

22.06 

94 

82 

61 

20 

2 

3 

0 

10 

98 

.39 

13 

26.02 

91 

70 

30 

3 

4 

0 

12 

12.02 

44 

20 

10 

29.01 

80 

40 

4 

31.5 

0.0 

6.14 

12.05 

17.50 

22.27 

26.19 

29.10 

30.90 

3I-5° 

31.5 

6 

o 

16 

09 

55 

35 

27 

19 

31.00 

60 

6 

7 

0 

18 

13 

61 

42 

35 

29 

10 

70 

7 

8 

o 

20 

17 

66 

49 

44 

38 

19 

80 

8 

9 

o 

22 

21 

72 

56 

47 

29 

90 

9 

32.O 

0.0 

6.24 

12.24 

17.78 

22.63 

26.61 

29.57 

31-39 

32.00 

32.0 

1 

o 

26 

28 

83 

70 

69 

66 

49 

10 

1 

2 

0 

28 

32 

89 

77 

77 

75 

59 

20 

2 

3 

0 

3° 

36 

94 

84 

85 

84 

68 

30 

3 

4 

0 

32 

40 

18.00 

94 

94 

78 

40 

4 

32.5 

0.0 

6.34 

12.44 

18.05 

22.98 

27.02 

30.03 

31.88 

32-50 

32.5 

6 

o 

36 

47 

ii 

23-05 

10 

12 

98 

60 

6 

7 

0 

38 

17 

13 

19 

22 

32.08 

7° 

7 

8 

o 

40 

55 

22 

20 

27 

31 

17 

80 

8 

9 

0 

42 

59 

28 

27 

35 

40 

27 

90 

9 

33.0 

0.0 

6.44 

12.63 

18.33 

23-34 

27.44 

30-49 

32.37 

33-0° 

33.0 

1 

0 

46 

67 

39 

4i 

52 

58 

47 

10 

1 

2 

o 

48 

7° 

44* 

48 

60 

67 

56 

20 

2 

3 

0 

5° 

74 

5° 

55 

68 

76 

66 

3° 

3 

4 

0 

52 

78 

56 

62 

77 

86 

76 

40 

4 

33.5 

0.0 

6-53 

12.82 

18.61 

23.69 

27.85 

30.95 

32.86 

33-50 

335 

6 

o 

55 

86 

66 

76 

93 

21.04 

95 

60 

6 

7 

o 

57 

90 

72 

83 

28.02 

13 

33-°5 

70 

7 

8 

o 

59 

93 

78 

90 

10 

23 

15 

80 

8 

9 

o 

61 

97 

83 

97 

19 

32 

25 

90 

9 

34.0 

0.0 

6.63 

13.01 

18.89 

24.04 

28.27 

3M1 

33-35 

34.00 

340 

1 

o 

65 

°5 

94 

12 

35 

51 

45 

10 

1 

2 

0 

67 

09 

19.00 

19 

43 

60 

20 

2 

3 
4 

0 

o 

69 

11 

05 
ii 

26 

33 

g 

69 

78 

74 

30 
40 

3 
4 

31.5 

o.o 

6-73 

13.20 

19.16 

24.40 

28.68 

3I-87 

33.84 

34-50 

31.5 

6 

o 

75 

24 

22 

47 

77 

97 

94 

60 

6 

7 

8 

0 

o 

77 
79 

32 

27 

33 

I? 

85 
93 

32.06 
15 

34.03 
13 

g 

7 

8 

» 

o 

81 

35 

39 

68 

29.01 

24 

23 

90 

9 

35.0 

o.o 

6.83 

13-39 

19.44 

24.75 

29.10 

32-34 

34-33 

35-00 

35.0 

1 

o 

85 

43 

5° 

82 

18 

43 

43 

10 

1 

2 
3 

o 

0 

87 
89 

47 
51 

55 
61 

89 
96 

26 
35 

£ 

g 

20 

3° 

2 
3 

4 

o 

54 

67 

25-03 

43 

7° 

72 

40 

4 

35.5 

0.0 

6.92 

I3-58 

19.72 

25.10 

29.51 

32.80 

34-82 

35-50 

35.5 

6 

o 

94 

62 

78 

18 

60 

89 

92 

60 

6 

7 

o 

96 

66 

83 

25 

68 

98 

35-02 

70 

7 

8 

o 

98 

70 

89 

32 

77 

33-o8 

12 

80 

8 

9 

o 

7.00 

74 

94 

39 

g5 

17 

21 

90 

9 

36.O 

0.0 

7.02 

13-78 

20.00 

25.46 

29-93 

33-26 

35.31 

36.00 

36.0 

TABLE  LIV. 


16' 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

! 

o.ooo 

0.195 

82  = 

0.383 

0.556 

0.707 

S.,= 

0.831 

0.924 

0.981 

1.000 

36.0 

o 

o.o 

o 

7.02 

I3°78 

0 

20.00 

2546 

29°93 

33°26 

0 

35-31 

36.00 

36°.O 

.  1 

o 

04 

8l 

05 

53 

30.01 

35 

41 

10 

1 

2 

o 

06 

85 

II 

60 

10 

44 

51 

20 

2 

3 

0 

08        89 

16 

67 

18 

54 

61 

3° 

3 

4 

o            10    |        93 

22 

74 

26 

63 

70 

40 

4 

36.5 

0.0 

7.12 

13-97 

20.28 

25.81 

30-35 

33-72 

35.80 

36.50 

36.5 

6 

o 

14 

14.00 

33 

88 

43 

81 

90 

60 

6 

7 

o 

16 

04 

39 

95 

91 

36.00 

7° 

7 

8 

o 

18 

08 

44 

26.02 

60 

34.00 

10 

80 

8 

9 

o 

20 

12 

50 

09 

68 

09 

19 

90 

9 

37.0 

o.o 

7.22 

I4.I6 

20-55 

26.16 

30.76 

34-18 

36.29 

37-00 

37.0 

1 

o 

24 

20 

61 

23 

85 

28 

39 

10 

1 

2 

0 

26 

23 

66 

3° 

93 

37 

49 

20 

2 

3 

o 

28 

27 

72 

38 

31.01 

46 

3° 

3 

4 

o 

3° 

31 

78 

45 

ib 

55 

69 

40 

4 

37.5 

o.o 

7.31 

14-35 

20.83 

26.52 

31.18 

34-64 

36.78 

37-50 

37.5 

6 

0 

33 

39 

89 

59 

26 

74 

88 

60 

6 

7 

o 

35 

42 

94 

66 

34 

83 

98 

70 

7 

8 

0 

37 

46 

21.00 

73 

43 

92 

37.08 

80 

8 

9 

o 

39 

5° 

°5 

80 

35-01 

17 

90 

9 

38.0 

0.0 

7.41 

14-54 

21.  II 

26.87 

31-59 

35-11 

37-27 

38.00 

38.O 

1 

o 

43 

58 

16 

94 

67 

20 

37 

10 

1 

2 

0 

45 

62 

22 

27.01 

76 

29 

47 

20 

2 

3 

o 

47 

66 

28 

08 

84 

38 

57 

30 

3 

4 

o 

49 

70 

33 

15 

92 

47 

66 

40 

4 

38.5 

0.0 

7-51 

H-73 

21-39 

27.22 

32.01 

35-57 

37-76 

38.50 

38.5 

6 

0 

53 

77 

44 

3° 

09 

86 

60 

6 

7 

o 

55 

81 

5° 

37 

17 

75 

96 

70 

7 

8 

6 

57 

85 

55 

44 

26 

84 

38.06 

80 

8 

9 

o 

59 

89 

61 

34 

94 

16 

90 

9 

39.0 

o.o 

7-6! 

14.92 

21.67 

27.58 

32.42 

36-03 

38.26 

39.00 

39.0 

1 

o 

63 

96 

72 

65 

51 

J3 

35 

10 

1 

2 

0 

65 

15.00 

78 

72 

59 

22 

45 

20 

2 

3 

o 

67 

04 

83 

79 

68 

31 

55 

3° 

3 

4 

0 

69 

08 

89 

86 

76 

40 

65 

40 

4 

39.5 

o.o 

7.71 

15-11 

21.94 

27-93 

32.84 

36.49 

38.74 

39-5° 

39.5 

6 

o 

73 

15 

22.00 

28.00 

92 

84 

60 

6 

7 

o 

75 

19 

°5 

07 

33-01 

68 

94 

70 

7 

8 

0 

77 

23 

ii 

15 

09 

77 

39-04 

80 

8 

9 

o 

79 

27 

17 

22 

17 

86 

*4 

90 

9 

4O.O 

o.o 

7.80 

I5-3I 

22.22 

28.29 

33-26 

36.95 

39.24 

40.00 

4O.O 

1 

2 

o 
o 

82 
84 

* 

27 

33 

36 

43 

34 

42 

37-04 
H 

33 
43 

10 

20 

1 
2 

3 
4 

o 

0 

86 
88 

42 
46 

39 
44 

50 
57 

.  59 

23 

33 

i 

30 
40 

3 
4 

40.5 

0.0 

7.90 

I5-50 

22.50 

28.64 

33-68 

37-42 

39-73 

40.50 

40.5 

6 

0 

92 

54 

55 

71 

76 

51 

82 

60 

6 

7 

8 

o 
o 

94 
96 

if 

61 
67 

78 
85 

84 
93 

60 

92 
40.02 

£ 

7 
8 

9 

o 

98 

65 

72 

92 

34-oi 

78 

12 

90 

9 

41.0 

0.0 

8.00 

15.69 

22.78 

28.99 

34-09 

37-88 

40.22 

41.00 

41.O 

1 

o 

02 

72 

83 

29.07 

17 

97 

31 

10 

1 

2 

o 

04 

76 

89 

14 

25 

38.06 

41 

20 

2 

3 

0 

06 

80 

95 

21 

34 

16 

51 

3° 

3 

4 

o 

08 

84 

23.00 

28 

42 

25 

61 

40 

4 

41.5 

o.o 

8.10 

15.88 

23.06 

29,35 

34-51 

38.34 

40.71 

41.50 

41.5  i 

6 

o 

12 

92    ii 

42 

59 

43 

80 

60 

6  ' 

7 

o 

13 

95     j        16 

49 

67 

52 

90 

70 

7 

8 

o 

15 

99            22 

56 

75 

62 

41.00 

80 

8 

9 

o 

17 

16.03            28 

63 

83 

71 

10 

90 

9 

42.O 

o.o 

8.19 

16.07 

23-33 

29.70 

34-92 

38.81 

41.19 

42.00 

42.0 

168 


TABLE  L1V. 


Products  of  Arcs  multiplied  by  the  Sines  of  the  Rhumbs. 

Arcs. 

Sines  of  the  Rhumbs. 

Arcs. 

S0  = 

o.ooo 

81  = 

0.195 

0.383 

0.556 

0.7O7 

0.831 

0.924 

£t      m  

O.981 

1.000 

o 

42.0 

o 

o.o 

8% 

o 

16.07 

23°33 

2970 

34°92 

38°8i 

o 

41.19 

o 

42.00 

42.O 

1 

o 

21 

II 

°39 

77 

35-0° 

90 

29 

10 

1 

2 

0 

23 

15 

44 

84 

08 

99 

39 

20 

2 

3 

o 

25 

19 

5° 

9' 

16 

39-oS 

49 

3° 

3 

4 

o 

27 

23 

56 

98 

25 

17 

59 

40 

4 

42.5 
6 

o.o 

0 

8.29 
31 

16.26 

3° 

23.61 
66 

30-05 

12 

35-34 
42 

39.26 
36 

41.69 
78 

42.50 
60 

42.5 
6 

7 

0 

33 

72 

19 

5° 

45 

88 

70 

7 

8 

o 

35 

38 

77 

26 

58 

54 

98 

80 

8 

9 

o 

37 

42 

83 

24 

67 

64 

42.08 

90 

9 

43.O 

o.o 

8-39 

16.46 

23.89 

30.41 

35-75 

39-73 

42.18 

43-0° 

43.0 

1 

o 

49 

94 

48 

83 

82 

28 

10 

1 

2 

o 

43 

53 

24.00 

55 

92 

91 

37 

20 

2 

3 

o 

45 

57 

05 

62 

36.00 

40.00 

47 

30 

3 

4 

o 

47 

pi 

ii 

69 

08 

10 

57 

40 

4 

43.5 
6 

0.0 

o 

8-49 

I6^ 

24.17 

22 

30.76 

36-  i  7 
25 

40.19 
28 

42.67 
77 

43-50 
60 

43.5 

6 

7 

0 

53 

72 

28 

90 

33 

37 

86 

70 

7 

8 

o 

55 

76 

33 

97 

42 

47 

96 

80 

8 

9 

o 

57 

80 

39 

31.04 

50 

56 

43-o6 

90 

9 

44.0 

o.o 

8.58 

16.84 

24-45 

31.11 

36.58 

40.65 

43.16 

44-00 

44.O 

1 

0 

60 

87 

5° 

18 

66 

74 

25 

10 

1 

2 

o 

62 

91 

55 

26 

75 

83 

35 

20 

2 

3 

o 

64 

95 

61 

33 

83 

93 

45 

30 

3 

4 

0 

66 

99 

67 

40 

92 

41.02 

55 

40 

4 

44.5 

0.0 

8.68 

17-03 

24.72 

31-47 

37-00 

41.11 

43-65 

44-50 

44.5 

6 

o 

70 

07 

77 

54 

08 

20 

74 

60 

6 

7 

o 

72 

10 

83 

pi 

16 

3° 

84 

70 

7 

8 

0 

74 

4 

89 

68 

25 

39 

94 

So 

8 

9 

o 

76 

18 

95 

75 

33 

48 

44.04 

90 

9 

45.O 

o.o 

8.78 

17.22 

25.00 

31.82 

37-41 

41-57 

44.14 

45.00 

45.O 

1 

0 

80 

26 

06 

89 

5° 

67 

24 

10 

1 

2 

3 

o 

0 

82 
84 

29 
33 

ii 
17 

96 
32-03 

58 

66 

c 

33 
43 

20 

3° 

2 
3 

4 

0 

86 

37 

22 

ii 

75 

95 

53 

40 

4 

45.5 

0.0 

8.88 

17.41 

25.28 

32.18 

37.83 

42.04 

44-63 

45-50 

45.5 

6 

o 

90 

45 

33 

25 

73 

60 

6 

7 

0 

o 

92 
94 

49 
53 

39 
45 

32 
39 

99 
38.08 

22 
32 

83 
93 

g 

7 
8 

9 

0 

95 

56 

50 

46 

16 

41 

45.02 

90 

9 

46.0 

0.0 

8.97 

17.60 

25-56 

32-53 

38.25 

42.50 

45-12 

46.00 

46.0 

1 

o 

99 

64 

Si 

60 

33 

59 

22 

IO 

1 

2 

0 

9.01 

68 

66 

67 

68 

31 

20 

2 

3 

o 

°3 

72 

72 

74 

49 

78 

41 

3° 

3 

4 

o 

05 

75 

78 

81 

57 

87 

5' 

40 

4 

46.5 

o.o 

9.07 

17.79 

25-84 

32.88 

38.66 

42.96 

45.61 

46.50 

46.5 

6 

0 

09 

83 

89 

95 

74 

43-05 

7i 

60 

6 

I 

o 

v  o 

ii 

87 
9i 

94 
26.00 

33-02 

10 

82 

»4 

24 

81 

g 

7 

8 

9 

0 

15 

95 

06 

17 

99 

33 

46.01 

90 

9 

47.0 

0.0 

9.17 

17.99 

26.11 

33-24 

39.08 

43-42 

46.10 

47.00 

47.0 

1 

0 

19 

18.02 

16 

31 

16 

51 

20 

10 

] 

2 

o 

21 

06 

22 

38 

24 

60 

29 

20 

2 

3 

o 

23 

10 

28 

45 

33 

70 

39 

3° 

3 

4 

0 

25 

14 

33 

52 

79 

49 

40 

4 

47.5 

0.0 

9.27 

18.18 

26.39 

33-59 

39-49 

43-88 

46.59 

47.50 

47.5 

6 

0 

29 

22 

44 

66 

57 

97 

69 

60 

6 

7 

o 

31 

25 

5° 

73 

66 

44.07 

/ 

70 

7 

8 

o 

32 

29 

55 

80 

74 

16 

88 

80 

8 

9 

o 

34 

33 

ll 

87 

82 

25 

98 

90 

9 

48.0 

o.o 

9-36 

18.37 

26.66 

33-94 

39-90 

44-34 

47.08 

48.00 

48.O 

TABLE  LV. 


169 


Magnetic  Element*  of  the  Earth:  The  Magnetic  Variation. 

IBf  ARCTIC  LATITUDES. 

T  »»*• 

Longitude  West  of  Greenwich. 

Lat. 

Juat. 

O° 

5° 

1O° 

15° 

20° 

25° 

30° 

35° 

40° 

45° 

60  N, 
!  65 
I  7O 
I  75 
|  SO 

60N 
65 
70 
75 
80 

60  N. 
65 
70 
75 
SO 

22  W. 

27 
28 

o 

25  W. 
28 
3° 
3i 
32 

30  w. 

33 

it 

37 

35  W. 
3» 

39 
40 

4i 

39  W. 
43 
44 
45 
45 

43  W. 

46 
48 
49 
50 

46  W. 
5° 
53 
54 
55 

49  W. 

it 

ii 

51  W. 

£ 

| 

52  w. 

61 

67 
69 
70 

60  N. 
65 
70 
75 

SO 

60  N. 
65 
70 
75 

SO 

60  N. 
65 
70 
75 
SO 

45° 

50° 

55° 

60° 

65° 

7O° 

75° 

80° 

85° 

90° 

52  W. 
61 

67 
69 
70 

54  W. 
64 

72 
75 
77 

Bw- 

& 
84 

53  W. 

O7 

91 

52  W. 

67 
81 
90 
98 

49  W. 
66 
82 
96 
105 

43  W. 
64 

83 
103 
114 

32  w. 

60 

84 

112 
I23 

20  W. 

5° 
85 
124 

132 

4W. 
25 
85 
135 
141 

90° 

95° 

100° 

105° 

11O° 

115° 

12O° 

125° 

13O° 

135° 

o 

4w. 

Si 

135 
I4I 

o 

loE. 
10  E. 
80  W. 
150 
150 

o 

19  E. 

70  E. 

I72\V. 
1  60 

o 

24  E. 

& 

156  E. 
I75W. 

o 

27  E. 

ii 

130 

I72E. 

0 

29  E. 
43 
65 
no 
1  60 

3?E. 
45 
63 
95 
140 

3°3E. 

45 
60 
88 
128 

33  E. 

44 

i 

no 

33  E. 

42 
54 

£ 

60  N. 
65 
70 
75 
SO 

135° 

140° 

145° 

150° 

155° 

160° 

165° 

170° 

175° 

ISO0 

33  E. 

42 

54 

^s 

3*E. 

40 

1° 
62 

84 

3?*. 
\ 

S^ 
70 

29  E. 

35 
42 

So 
59 

0 

27  E. 
33 
39 
45 
54 

o 

25  E. 

30 
35 

40 

49 

o 

23  E. 

27 
32 
37 
45 

o 
21  E. 

24 
29 

35 
42 

i8E. 

21 

26 

32 
40 

o 

i6E. 
19 
23 
30 

38 

60  N. 
65 
70 
75 
SO 

Lftt 

Longitude  East  of  Greenwich. 

Lat. 

0° 

5° 

1O° 

15° 

2O° 

25° 

3O° 

35° 

40° 

45° 

60  N. 
65 
70 
75 
SO 

60  N. 
65 
70 
75 
SO 

60  N. 
65 
70 
75 
SO 

60  N. 
65 
70 
75 
SO 

22  W. 

24 
26 
27 
28 

I9W. 

20 
22 

22 
23 

i5W. 

18 
18 
19 

12  W. 
14 

J3 
13 
'3 

r- 

9 

10 

10 

6W. 

5 

6 

2W. 

2 

2 
2 

»i 

2 
2 
2 
2 

0 

4E. 

1 

6 

7 

o 

6E. 
8 
9 
9 

10 

60  N. 
65 
70 
75 

80 

60  N. 
65 
70 
75 
80 

45° 

50° 

55° 

60° 

65° 

7O° 

75° 

8O° 

85° 

9O° 

o 

6E. 
8 
9 
9 

10 

8E. 

10 
12 
13 
13 

o 

10  E. 

12 

H 
15 
15 

i?E. 

11 

18 
18 

12  E. 

16 
18 

21 

22 

0 

12  E. 

16 
18 

24 
25 

13  E. 
10 

18 

1 

o 
12  E. 

16 
18 
25 
3i 

o 

ii  E. 
H 
17 

24 

33 

o 

10  E. 
13 
15 

22 
32 

9O° 

95° 

10O° 

105° 

110° 

115° 

12O° 

125° 

130° 

135° 

o 

loE. 
'3 
15 

22 
32 

°7E. 
9 
13 

20 
31 

SB. 
7 
ii 
18 
30 

SB. 

5 
9 
16 

29 

o 
0 

3E. 

7 
15 

28 

o 
2W. 
0 

6E. 

H 
27 

o 

3W. 

i 
5E. 

13 
27 

o 

sw. 

2 

4E. 
13 

27 

o 

6W. 

?*: 

12 

27 

o 

5W. 

U 

12 

27 

60  N. 
65 
70 
75 
SO 

60  N. 
65 
70 
75 
SO 

135° 

140° 

145° 

15O= 

155° 

160° 

165° 

170° 

175° 

ISO0 

o 

5W. 

U 

12 

27 

o 

4W. 
i 

4E. 

13 

27 

2W. 

IE. 

6 

14 

28 

o 

o 

2E. 

8 
16 
29 

ffc 

10 

18 
30 

\E- 

12 
2O 
31 

KF 

0  EJ. 

10 

i"S 

22 
32 

o 

10  E. 
13 
17 

24 

34 

;iE- 

20 

I 

i6E. 
19 
23 
30 
38 

170 


TABLE  LVI. 


i 

magnetic  Elements  of  the  Earth  :  The  magnetic  Variation. 

IN  LATITUDES  FRO  11  70°  X.  TO  6O<>  S. 

Longitude  West  of  Greenwich. 

Lat. 

Lat. 

o° 

5° 

10° 

15° 

20° 

25° 

30° 

35° 

40° 

45° 

70  N. 
65 

25°7W. 
23-7 

30?2W. 
28.0 

35-oW. 
32.8 

39°5W. 
38.0 

44^2  W. 
42.6 

48?6W. 
46.6 

50-5 

57°5W. 
54-5 

58.5^ 

67.5  W. 
61.0 

7ON. 
65 

i  60 

22.0 

254 

30.0 

34-8 

394 

43-3 

46.5 

49.0 

5°-3 

52-6 

6O 

55 

20.6 

234 

27.0 

31.3 

43-2 

44-3 

44-8 

55 

50 

19-3 

21.8 

244 

27-5 

3o!8 

33-5 

35-5 

37-o 

37-8 

37-6 

50 

45  N. 

i8.2W. 

20.3  W. 

22.6  W. 

24.9  W. 

27.4W. 

29.8  W, 

31-3  w- 

32.  oW. 

31.9  W. 

3i.o\V. 

45  N. 

40 
35 

\u 

19.1 

18.2 

21.0 

19.8 

23.0 
21.4 

24.7 
23.8 

26.4 
24.0 

27-3 
24.2 

27-5 
23.8 

27.0      j  25.5 
23.0       21.4 

40 
35 

30 

16.4 

17.6 

1  8.9 

20.2 

21.2 

21.7 

21.5 

20.8 

19-5 

17.0 

30 

25 

16.4 

17.4 

18.4 

194 

2O.2 

20.3 

19.9 

18.8 

16.5 

13.8 

25 

i  2O  N. 

i6.6W. 

I7-5W. 

18.2  W. 

19.0  W. 

19-5  W. 

19.5  W. 

i8.7\V. 

16.5  W. 

i4.o\V. 

io.7\V. 

20  N. 

15 

17.1 

17-7 

18.3 

I9.O 

194 

19.2 

17-3 

15.0 

1  1.6 

8.2 

15 

10 

17.6 

18.2 

18.6 

19.2 

19-5 

18.7 

16.4 

9.8 

6.0 

10 

5 

18.4 

18.8 

19.3 

19-5 

19.4 

18.2 

15.8 

12.5 

8.8 

5-° 

5 

0 

19.8 

20.5 

20.7 

20.3 

19-5 

18.9 

15-3 

12.0 

8-3 

44 

O 

58. 
10 

21.8  W. 
24.0 

22.4  W. 
24.2 

22.5  W. 
23.8 

2I.6W. 

22.O 

i9-9\V. 
19.6 

I7.5W. 
17.0 

15-oW. 
H-3 

II.O 

8.0  W. 

74 

4.0  W. 

3-6 

58.  ! 
10 

15 

25-8 

25.6 

24.2 

21.6 

18.8 

16.2 

13-5 

10.4 

6.6 

3-° 

15 

20 

27.0 

25.8 

234 

20.7 

1  8.0 

15-3 

12.5 

9.2 

54 

2.2 

20 

25 

26.9 

25.0 

22.3 

19.6 

1  6.8 

14.1 

1  1.2 

7-7 

4-3 

1.2 

25 

308. 

26.0  W. 

23-7  w. 

21.  1  W. 

18.4  W. 

I5.6W. 

i2.7\V. 

9.6  W. 

6.4  W. 

3.0  W. 

O.O 

308. 

35 

24.8 

22.4 

19.8 

17.0 

H-3 

II.2 

8.2 

5-° 

1.6 

i.9E. 

35 

4O 

234 

21.0 

18.4 

15.7 

12.8 

9.8 

6.6 

34 

O.O 

3-5 

40 

45 

21.9 

19-5 

16.8 

14.2 

II.  2 

8.0 

5-° 

2.0  E. 

5-3 

45 

50 

20.3 

17-7 

15.2 

12.4 

9-5 

6-3 

3-° 

0.3  E. 

3-7 

7.0 

5O 

558. 

18.5  W. 

I5-9W. 

13.3  W. 

10.5  W. 

74  W. 

4.2  W. 

0.8  W. 

2.4  E. 

5.6  E. 

8.6  E. 

55  S. 

60 

16.8 

14.0 

11.4 

8.3 

5-2 

2.8 

1.2  E. 

44 

7-3 

10.3 

6O 

Longitude  West  of  Greenwich. 

Lat. 

Lat. 

45° 

5O° 

55° 

60° 

65° 

70° 

75° 

80° 

85° 

90° 

70  N. 

67^5  w. 

7i°5W 

75°oW. 

78?o  W. 

8i?oW. 

82?oW. 

83.0  W. 

84  W. 

85^0  W. 

84?o\V. 

70  N. 

65 

61.0 

64*5 

66.5 

67.0 

67.5 

66.5 

64-5 

60.0 

50.0 

22.5 

65 

60 

52.6 

53-8 

54.0 

53-3 

44.0 

42-5 

32.0 

2O.O 

3-5 

60 

55 

44-8 

44.8 

43-8 

42.0 

38^0 

33-° 

27.0 

1  8.0 

8.0 

3-0  E. 

55 

50 

37-6 

36-3 

34-2 

31.0 

27.0 

22.4 

16.5 

I  O.O 

2.2 

6.0 

50 

45  N. 

3i.oW. 

29.3  W. 

26.8  W. 

23.7  W. 

20.3  W. 

iS.oW. 

10.0  W. 

4.5  W. 

i.oE. 

7.0  E. 

45  N. 

40 

25-5 

23-5 

20.9 

17-5 

134 

94 

5-3 

I.O 

3-3 

7-3 

4O 

35 

21.4 

19.0 

15-7 

12.3 

9.0 

5-8 

2-3 

i.i  E. 

44 

74 

35 

30 

17.0 

14.0 

n.  i 

8.5 

5-7 

2-7 

0.0 

2.6 

5-° 

7-3 

30 

25 

13.8 

10.9 

8.4 

5-8 

1.6  E. 

3-5. 

54 

7.2 

25 

20  N. 

10.7  W. 

7.0  W. 

5.2  W. 

2.7\V. 

o.iW. 

1.4  E. 

2.8  E. 

4.4  E. 

5.9  E. 

7-3  K. 

2O  N. 

|  15 

8.2 

5-2 

2.4 

0.2  E. 

1.4  E. 

2-5 

3-8 

5-2 

6.5 

74 

15 

10 

6.0 

3-° 

o.i  E. 

1.4 

2-5 

3-7 

4-9 

6-3 

7.2 

7-7 

10 

5 

5-° 

1.8 

0.8 

2.1 

3-3 

4.6 

6.0 

7-i 

7-8 

8-3 

5 

O 

44 

I.O 

1.3 

2.7 

4.0 

5.6 

7.0 

8.0 

8.6 

8.7 

O 

58. 

4.0  W. 

0.6  W. 

1.7  E. 

3-3  E. 

5.0  E. 

6.5  E. 

8.0  E. 

9.0  E. 

9.4  E. 

9.5  E. 

58. 

10 

3-6 

0.2 

2.1 

4.2 

5.8 

7-7 

9.0 

IO.I 

10.5 

10.5 

1O 

15 

3-° 

0.2  E. 

2.6 

5.0 

7.0 

9.0 

10.2 

1  1.2 

"•5 

11.5 

15 

20 

2.2 

I.O 

3-7 

6  3 

8-5 

10.3 

u.6 

12.5 

12.8 

12.6 

20 

25 

1.2 

2.3 

7^8 

1  0.0 

1  1.  8 

12.2 

I4.2 

14-3 

13.8 

25 

308. 

O.O 

3-5  E. 

6.5  E. 

9.4  E. 

ii.6E. 

13.8  E. 

15.2  E. 

15.8  E. 

15.8  E. 

15.1  E. 

308. 

35 

1.9  E. 

S-2 

8-3 

II.O 

J3-5 

I5-5 

1  6.8 

17.4 

17.2 

16.4 

35 

40 

3-5 

6.8 

10.0 

12.8 

154 

i7-3 

18.5 

19.1 

19.0 

18.2 

4O 

45 

5-3 

8.4 

1  1.6 

14.7 

19.0 

20.4 

21.2 

21.  1 

20.4 

45 

50 

7.0 

I  O.O 

13-3 

16.1 

18.3 

20.5 

22.1 

23.2 

23.2 

22.7 

5O 

558. 

8.6  E. 

ii.8E. 

15.0  E. 

I7-5E. 

20.0  E. 

21.8  E. 

24.0  E. 

2t;.oE. 

25.5  E. 

25-3  E. 

558. 

60 

10.3 

13-5 

16.5 

19.2 

21.6 

24-3 

26.O 

27.1 

27.7 

27.8 

6O 

TABLE  LVI. 


171 


Magnetic  Elements  of  the  Earth  :   The  Magnetic  Variation. 

IN  LATITUDES  FROM  7O    X.  TO  60    S. 

Longitude  West  of  Greenwich. 

Lat. 

Lat. 

90° 

95° 

I  OO° 

105° 

1100 

115° 

120° 

125° 

130° 

135° 

70  N. 

84.0  W. 

8o.oW. 

o 

70.0  E. 

68.0  E. 

67.0  E. 

65°.o  E. 

63.0  E. 

60.0  E. 

58.0  E. 

54.0  E. 

7ON. 

65 

22.5 

7-5  E. 

23.0 

31.5 

38.0 

42.6 

45  -° 

45-o 

44-0 

42.0 

65 

60 

3-5 

IO.O 

19.0 

24.0 

27.2 

29-5 

31.0 

33-o 

33-o 

32.6 

60 

55 

3.0  E. 

II.O 

17.0 

2I.O 

23.1 

25.0 

26.3 

27.2 

27-5 

27.4 

55 

5O 

6.0 

10.8 

15-3 

l8.4 

21.6 

22.0 

22.8 

23-4 

23-7 

23-6 

50 

45  N. 

7.0  E. 

10.4  E. 

13.4  E. 

i6.oE. 

I7.8E. 

19.3  E. 

20.0  E. 

20.4  E. 

20.5  E. 

20.4  E. 

45  N. 

4O 

7-3 

IO.O 

12.0 

14.0 

15-4 

16.4 

17.0 

17-5 

17.7 

17.7 

40 

35 

7-4 

9-3 

II.O 

12.5 

13-5 

14.3 

14.9 

15.2 

15-4 

15-4 

35 

30 

7-3 

8.9 

IO.O 

II.O 

11.7 

12.2 

12.5 

12.7 

12.8 

12.9 

30 

25 

7-2 

8-4 

9-1 

9.8 

10.3 

IO.6 

10.7 

10.8 

10.8 

10.8 

25 

20  N. 

7-3  E 

8.1  E. 

8.5  E. 

8.8  E. 

9.0  E. 

9.0  E. 

9.0  E. 

9.0  E. 

8.9  E. 

8.9  E. 

20  N. 

15 

7-4 

8.0 

8-3 

8.1 

7-9 

7.6 

7-3 

7.1 

7.0 

7.0 

15 

1O 

7-7 

8.1 

7-9 

7-3 

6.9 

6.5 

6.2 

6.0 

5.8 

5.8 

10 

5 

8.3 

8-3 

7-7 

7.0 

6.4 

6.0 

5-5 

5-0 

4.8 

4.8 

5 

0 

8-7 

8.6 

8.0 

6-3 

5-7 

4-5 

4.0 

4.0 

0 

58. 

9-5  E. 

9.2  E. 

8.4  E. 

7.5  E. 

6.6  E. 

6.0  E. 

5-4E. 

5.0  E. 

4.5  E. 

4.4  E. 

58. 

10 

10.5 

IO.O 

9.1 

8.1 

7.2 

6.6 

6.0 

5-7 

5-5 

5-5 

10 

15 

10.9 

9.9 

9.0 

8.0 

7-4 

7.0 

6.6 

6.4 

6-3 

15 

20 

12.6 

11.8 

10.8 

9.9 

8.9 

8.2 

7-8 

7-5 

7-3 

7.2 

20 

25 

13-8 

12.9 

11.7 

10.7 

9.8 

9.0 

8.6     . 

8-3 

8.2 

8.1 

25 

SOS, 

15.1  E. 

14.0  E. 

13.0  E. 

II.8E. 

10.7  E. 

9.8  E. 

9-3  E. 

9.0  E. 

8.7  E. 

8.7E. 

308. 

35 

16.4 

15-4 

14.1 

13.0 

11.9 

10.9 

IO.I 

9-7 

9-5 

9.4 

35 

40 

18.2 

17.0 

15.6 

14.4 

13.2 

12.2 

"•3 

10.6 

10.3 

IO.I 

40 

45 

20.4 

17.8 

16.2 

14.6 

I3.6 

12.7 

I2.O 

"•5 

1  1.2 

45 

50 

22.7 

21.6 

20.2 

18.6 

17.0 

15-3 

14.2 

13-5 

13-0 

12.6 

50 

558. 

25.3  E. 

24.5  E. 

23.0  E. 

2I.4E. 

19.8  E. 

18.2  E. 

I6.6E. 

15-5  E. 

14.8  E. 

14.4  E. 

558. 

60 

27.8 

27-5 

26.7 

25.2 

23-6 

22.0 

20.4 

19.1 

18.1 

17-5 

60 

Longitude  West  of  Greenwich. 

Lat. 

Lat. 

135° 

140° 

145° 

150° 

155° 

160° 

165° 

170' 

175° 

ISO0 

70  N. 

54.0  E. 

0 

50.0  E. 

46.5  E. 

0 

42.0  E. 

39.0  E. 

35  -°E. 

0 

29.3  E. 

25.8  E. 

23.0  E. 

70  N. 

65 

42.0 

40.0 

37-0 

34-5 

32.5 

30.0 

26^6  / 

23-9 

2I.O 

18.5 

65 

60 

32-6 

32.0 

30-7 

28.7 

26.7 

24.8 

22.9 

20.7 

l8.3 

16.0 

60 

55 

27.4 

26.8 

26.0 

24.8 

23-5 

22.0 

20.6 

18.7 

1  6.8 

14.7 

55 

50 

23.6 

23.2 

22.6 

21.9 

21.0 

19.9 

18.5 

17.0 

15.6 

14.0 

5O 

45  N. 

20.4  E. 

2O.O  E. 

19.8  E. 

19.3  E. 

i8.7E. 

iS.oE. 

i6.oE. 

15.8  E. 

14.7  E. 

13-5  E. 

45  N. 

40 

17.7 

17.6 

17-3 

17.0 

1  6.6 

1  6.0 

15-4 

14.7 

14.0 

13.0 

40 

35 

15-4 

15-3 

15.0 

14.9 

14.7 

14.4 

14-3 

13-9 

13-3 

12.4 

35 

30 

12.9 

12.9 

13.0 

13.0 

13.0 

13.0 

13.0 

12.9 

12.6 

12.0 

30 

25 

10.8 

10.9 

II.O 

11.  i 

"•5 

11.7 

1  1.  8 

11.9 

u.6 

25 

20  N. 

8.9  E. 

8.9  E. 

8.9  E. 

9.0  E. 

9-3-E. 

9.7  E. 

lo.iE. 

10.6  E. 

10.8  E. 

10.8  E. 

20  N. 

15 

7.0 

7.1 

7-3 

7-5 

8.0 

8.4 

9.0 

9-5 

9.8 

IO.O 

15 

1O 

5.8 

5-9 

6.t 

6.4 

6.8 

7-3 

8.0 

8.7 

9.1 

9-5 

10 

5 

4.8 

4.8 

5<o 

5-6 

6.2 

6-7 

7-4 

8.1 

8.6 

9.0 

5 

O 

4.0 

4.0 

4.2 

5-2 

5-9 

6.5 

7-2 

7-9 

8-3 

8.7 

0 

53. 

4.4  E. 

4.5  E. 

4.8  E. 

5-4  E. 

6.0  E. 

6.6  E. 

7-3  E. 

8.0  E. 

8.4  E. 

8.8  E. 

58. 

10 

5-5 

5-5 

5-7 

6.0 

6-5 

7  * 

7.6 

8-3 

8.8 

9.2 

10 

15 

6-3 

6-3 

6.4 

6-7 

7-2 

7.6 

8.2 

8.7 

9-3 

9.6 

15 

2O 

7-2 

7.2 

7-3 

7.6 

7.8 

8.4 

8.9 

9.4 

9.8 

10.2 

20 

25 

8.1 

8.1 

8.2 

8.4 

8.7 

9.2 

9.6 

IO.O 

10.7 

II.  I 

25 

3OS. 

8.7E. 

8.7E. 

8.8  E. 

9.2  E. 

9-5  E. 

9.8  E. 

10.3  E. 

10.9  E. 

n.6E. 

12.3  E. 

308. 

35 

9-4 

9-4 

9.6 

9-8 

IO.2 

10.7 

"•3 

I2.O 

12.7 

13-4 

35 

40 

IO.I 

IO.2 

10.4 

10.7 

11.2 

11.7 

12.4 

13.0 

13-8 

14.6 

4O 

45 
5O 

II.2 
12.6 

II.2 
12.7 

"•5 

12.8 

u.8 
13.2 

12.3 
13.6 

12.9 
14.2 

13-5 
15.0 

14.3 
15.8 

'5-2 

16.7 

16.0 
17-5 

45 

50 

558. 

14.4  E. 

14.4  E. 

14.5  E. 

15.0  E. 

15.4  E. 

I6.2E. 

i7.oE. 

I7.8E. 

18.4  E. 

19.0  E. 

558. 

6O 

17.5 

17-3 

17.4 

17.7 

18.2  , 

18.7 

19.2 

19.7 

20.3 

21.2 

6O 

172 


TABLE  LVI. 


magnetic  Elements  of  the  Earth:  The  Magnetic  Variation. 

IN  LATITUDES  FROM  7<P  BT.  TO  6O°  S. 

Longitude  East  of  Greenwich. 

Lat. 

Lat. 

0° 

5° 

10° 

15° 

20° 

25° 

30° 

35° 

10 

45° 

70  N. 

25.7  W. 

2I.8W. 

I7.6W. 

13.5  W. 

9.5  W. 

5.0  W. 

I.2W. 

o 

2.4  E. 

5.6  E. 

9°.o  E. 

70  N. 

65 

23-7 

2O.O 

16.4 

12.6 

8.9 

5-3 

1.6 

2.O 

4.9 

7-7 

65 

60 

22.O 

1  8.8 

15.4 

12.0 

8.6 

5-7 

2.4 

I.O 

3-7 

6.1 

60 

55 

20.6 

17.7 

14-5 

"•5 

8.5 

6.0 

3-2 

0.4  W. 

2.0 

4-3 

55 

50 

19.3      16.7 

13-8 

"•3 

8.6 

6-3 

4.0 

i-7 

°-3 

2-3 

50 

45  N. 

I8.2W.  !i6.oW. 

13.5  W. 

ii.3W. 

8.8  W. 

6.7W. 

4.6  W. 

2.5W. 

i.oW. 

0.8  E. 

45  N. 

40 

17-3       !  15-3 

13-3 

11.4 

9.2 

7.2 

5-o 

3-4 

1.8 

o.5\V. 

40 

35 

1  6.6 

15.0 

13.2 

11.5 

9.6 

7.6 

5.6 

3-9 

2-5 

i-3 

35 

30 

16.4 

15.0 

!3-3 

11.7 

1  0.0 

8.1 

6.2 

4-4 

3-1 

i-9 

30 

25 

16.4 

15* 

13-7 

I2.O 

10.3 

8.6 

6.7 

5-o 

3-7 

24 

25 

20  N. 

i6.6VV. 

I5.6W. 

14.2  W. 

I2.7W. 

n.oW. 

9.2  W. 

7.3  W. 

5.6W. 

4-2W. 

3.oW. 

20  N. 

15 

17.1 

16.2 

15.0 

13-4 

11.7 

I  O.O 

8.2 

6.4 

4.8 

3-6 

15 

10 

17.6 

16.9 

15-9 

14-5 

12.8 

1  1.0 

9.2 

7-4 

6.0 

44 

10 

5 

18.4 

17.6 

16.7 

14.0 

12.3 

10.4 

8.6 

6.8 

5-3 

5 

0 

19.8 

18.9 

17.9 

16.8 

15-5 

13-9 

I2.O 

I  O.O 

8.2 

6-5 

O 

ft  8. 

2I.8VV. 

20.8  W. 

19.5  W. 

i8.2\V. 

i7.o\V. 

I5.6W. 

14.0  W. 

I2.0W. 

9.8W. 

8.0  W. 

5S. 

10 

24.0 

23.2 

22.O 

20.3 

19.0 

17.4 

15.8 

14.0 

1  1.8 

9.8 

1O 

15 

25-8 

25-4 

24.6 

23-5 

21.7 

20.0 

18.2 

16.2 

14.4 

12.2 

15 

20 

27.0 

27.1 

26.8 

26.0 

24.6 

23.0 

21.0 

19-3 

17-3 

I5.O 

20 

25 

26.9 

28.1 

28.2 

27.9 

27.0 

25.7 

24.2 

22.5 

20.7 

18.8 

25 

3OS. 

26.0  W. 

27.8  W. 

28.8  W. 

29.  i  W. 

28.9  W. 

28.3  W. 

27.o\V. 

25.4W. 

23.7  W. 

22.0  W. 

308. 

35 

24.8 

26.8 

28.6 

29:6 

30.2 

30.2 

29.8 

28.9 

27-5 

26.O 

35 

40 

234 

25-7 

27.7 

29.5 

3°-9 

31.6 

3i-7 

3r-3 

30.6 

29.6 

40 

45 

21.9 

24.4 

26.6 

28.8 

30.8 

32.2 

33-2 

33-4 

33-o 

324 

45 

50  ' 

20.3 

23.0 

25-3 

27.8 

30.2 

32.3 

33-8 

34-9 

35-2 

35-2 

50 

558. 

18.5  W. 

2I.2W. 

23.8  W. 

26.4  W. 

29.2W. 

32.o\V. 

34-2W. 

35-8W. 

36.8  W. 

374W. 

558. 

60 

16.8 

19.7 

22.3 

25.0 

28.1 

3*-3 

34-o 

36.0 

37-3 

384 

6O 

Longitude  Bast  of  Greenwich. 

Lat. 

Lat. 

45° 

50° 

55° 

60° 

65° 

7O° 

75° 

80° 

85° 

90° 

70  N. 

0 

9.0  E. 

I2.0E. 

14.0  E. 

i6.oE. 

iS.oE. 

iS.oE. 

iS.oE. 

iS.oE. 

0 

i7.oE. 

15.0  E. 

70  N. 

65 

7-7 

10.  1 

12.5 

14.0 

16.0 

16.5 

16.5 

16.0 

14.0 

13.0 

65 

60 

6.1 

8.2 

10.0 

"•3 

12.0 

12.5 

12.6 

12.5 

"•3 

9-7 

6O 

55 

4-3 

6.2 

7-9 

9.1 

9-7 

1  0.0 

I  O.O 

Q.Q 

9-4 

8.0 

55 

50 

2-3 

4.0 

5-7 

5-9 

7.8 

8.3 

8.6 

8.5 

8.2 

7-i 

5O 

45  N. 

0.8  E. 

2.4  E. 

3-9  E. 

5.0  E. 

6.2  E. 

6.8  E. 

7.2  E. 

7.4  E. 

74  E. 

6.7  E. 

45  N. 

40 

0.5  W. 

0.9 

2.2 

3-2 

4.2 

5-2 

6.0 

6-3 

6.4 

6-3 

40 

35 

*-3 

0.2W. 

0.9 

1.8 

2.7 

3-5 

4.2 

4-7 

5-i 

5-2 

35 

I  30 

1.9 

I.I 

0.2  W. 

0.6 

1.4 

2.0 

2.6 

3-1 

3-5 

3-6 

30 

1  25 

2.4 

1.6 

0.8 

o.o 

0.7 

I.I 

i-7 

2.2 

2.7 

2.8 

25 

20  N. 

3.0  VV. 

2.0W. 

i.3\V. 

0.5  W. 

0.2  E. 

o.7E. 

1.2  E. 

1.6  E. 

2.1  E. 

2.3  E. 

20  N 

15 

3-6 

2-5 

1.6 

0.9 

o.o 

0.4 

0.8 

i-3 

1.6 

2.0 

15 

10 

4.4 

3-2 

2.1 

0.5  W. 

0.2 

0.6 

I.O 

i-3 

1.6 

10 

5 

5-3 

4.0 

2.8 

1.8 

I.O 

0.2W. 

0.4 

0.7 

i.i 

1.4 

5 

0 

6-5 

5-o 

3.8 

2.7 

1.7 

0.8 

o.o 

0.4 

0.7 

0 

ft  a. 

8.0  W. 

6.5  W. 

4.9W. 

3.7W. 

2.7W. 

i.8W. 

0.9  W. 

0.0 

0.3  E. 

0.5  E. 

58. 

10 

9.8 

8.1 

6.5 

5-° 

4.0 

3-° 

2.1 

i.4\V. 

o.7\V. 

0.2  W. 

10 

15 

12.2 

1  0.0 

8.5 

7-i 

3-8 

4-7 

3-9 

3-2 

2.6 

2.1 

15 

20 

15.0 

13.0 

II.O 

9-5 

8.2 

7-i 

6.2 

5-5 

4-9 

4-5 

20 

25 

ill 

15.6 

14.7 

13.0 

"•5 

10.3 

9.6 

9.0 

8.4 

7-7 

25 

30  S 

22.0  W. 

20.2  W. 

i8.5\V. 

i6.8W. 

i5.3W. 

14.2  W. 

13.0  W. 

I2.4W. 

I2.0W. 

u.S\V. 

308. 

35 

26.O 

24-3 

22.8 

21.5 

20.2 

19.1 

18.1 

'7-5 

17.0 

16.0 

35 

40 

29.6 

28.2 

27.0 

25.8 

24.7 

23-7 

23.0 

22.4 

21.6 

20.5 

4O 

45 

32.4 

31-7 

30.8 

30.0 

29.2 

28.4 

27.7 

27.2 

26.7 

25-7 

45 

50 

35-2 

34-7 

34-2 

33-7 

33-3 

32-8 

32-5 

32.2 

32.0 

31.2 

5O 

55  S. 

374W. 

37-7  W. 

38.0  W. 

38.2  W. 

38.5  w. 

39.0  W.:   39.3  W. 

39-5  W. 

39-2  W. 

38.3  w. 

558. 

60 

38-4 

39-3 

40.0 

41.7 

43-5 

45.0             1  46.5 

47.0 

46.2      1  45.0 

6O 

TABLE  LVI. 


173 


]fl  agnetic  Elements  of  the  E:irt£i  :   TiSi«  3I:i£fcieUc  Variation. 

FROM  LATITUDES  7O°  X.  TO  60°  S. 

Longitude  East  of  Greenwich. 

Lat. 

Lat. 

9O° 

95° 

100° 

105° 

11O° 

115° 

12O° 

125° 

130° 

135° 

70  N. 

0 

15.0  E. 

o 

13.0  E. 

i  i?o  E. 

9.0  E. 

0 

7.0  E. 

o 

5-5  E. 

o 

4-5  E. 

o 

3-5  E. 

0 

2.7  E. 

o 

3-5  E. 

70  N.  | 

65 

13.0 

9-4 

7.0 

5-° 

2-5 

o.o 

1.2  W. 

2.5  W. 

3.0  W. 

2.7W. 

65 

6O 

9-7 

7-3 

5-o 

2-5 

o.o 

1.6  W. 

3-4 

5-2 

6.0 

5.2 

6f> 

55 

8.0 

6.0 

3-8 

•4 

0.7  W. 

2.4 

4.1 

5-7 

6.5 

6.0 

55 

50 

7-i 

5-4 

3-3 

.0 

0.8 

2-5 

4-2 

5.8 

6.5 

6.1 

50 

45  N. 

6.7E. 

5.0  E. 

3.0  E. 

.oE. 

o.7W. 

2.2  W. 

3-9  W. 

5.6  W. 

6.4  W. 

6.0  W. 

45  N. 

40 

6-3 

4.8 

3-° 

.2 

o-3 

i-7 

3-3 

4-8 

5-7 

5-5 

40 

35 

5-2 

4-3 

3-i 

•4 

0.2 

1.2 

2-5 

3-9 

4-5 

4-4 

35 

30 

3-6 

3-4 

2.7 

•7 

0.6  E. 

0.6 

i-7 

2.6 

3-° 

2.9 

30 

25 

2.8 

2.9 

2-5 

2.0 

I.O 

o.i  E. 

0.8 

1.6 

1.8 

1.6 

25 

20  N. 

2-3  E. 

2.5  E. 

2.3  E. 

2.2  E. 

1.5  E. 

0.6  E. 

o.o 

o.7\V. 

0.9  W. 

0.5  W. 

20  N. 

15 

2.O 

2.2 

2.4 

2.2 

i-7 

1.2 

0.6  E. 

o.o 

o.o 

0.5  E. 

15 

10 

1.6 

1.9 

2.1 

2.O 

1.6 

1.4 

i-3 

.iE. 

i.i  E. 

i-4 

10 

5 

1.4 

1.6 

i.i 

i-7 

i-7 

1.6 

1.4 

•4 

1.5 

1.9 

5 

O 

i.i 

i-3 

1.4 

1.4 

1.4 

1.4 

i-5 

•7 

1.8 

2-3 

O 

5S. 

0.5  E. 

0.8  E. 

i.oE. 

i.iE. 

i.iE. 

1.2  E. 

1.3  E. 

.6E. 

1.8  E. 

2.7  E. 

53. 

1O 

0.2  W. 

O.I 

0.2 

°-3 

0.4 

°-5 

I.O 

•4 

i-9 

3-° 

1O 

15 

2.1 

i.7\V. 

i.4W. 

i.iW. 

0.8  W. 

0.5  W. 

0.0 

0.8 

i-7 

3-1 

15 

20 

4-5 

4.0 

3-4 

2.8 

2.2 

1.5 

0.8  W.    o.o 

1.4 

3-2 

2O 

25 

7-7 

7.0 

6.0 

4.9 

3-8 

2.6 

i-7 

0.6  W. 

0.9 

3-3 

25 

3OS. 

n.5\V. 

10.6  W. 

9.3  W. 

7.7  w. 

6.0  W. 

4.3  W. 

2.5  W. 

i.i  W. 

0.5  E. 

3-3  E. 

3OS 

35 

16.0 

14.6, 

13.0 

II.O 

9.0 

6.5 

4-3 

1.8 

o.o 

3-2 

35 

40 

20.5 

19.0 

16.8 

14.4 

1  1.6 

9.0 

6.0 

3-° 

0.3  W. 

3-° 

40 

45 

25-7 

24.0 

21.4 

18.4 

15-3 

11.9 

8-3 

4.8 

I.O 

2.8 

45 

50 

31.2 

29-5 

27.0 

23.0 

19.0 

15.0 

10.7 

6.5 

2.O 

2.5 

50 

558. 

38.3  w. 

35-8  W. 

33-oW. 

29.0  W. 

24.3  W. 

19.8  W. 

14.7  W. 

9.2  W. 

3.8  W. 

1.8  E. 

55  S. 

6O 

45-o 

43-2 

41.0 

36.0 

31.0 

25.0 

2O.O 

13-8 

7.0 

o.o 

60 

Longitude  East  of  Green  wich. 

Lat. 

Lat. 

135° 

140° 

145° 

150° 

155° 

160° 

165° 

170> 

175° 

18O° 

70  N. 

0 

3-5  E. 

o 

4-5  E. 

6.0  E. 

8?oE. 

10.0  E. 

o 

12.  5  E. 

K.oE. 

o 

17.5  E. 

20.0  E. 

o 

23.0  E. 

70  N. 

65 

2.7\V. 

I.4VV. 

°'5 

2-5 

4.6 

7-3 

9.8 

12.6 

15.8 

18.5 

65 

6O 

5-2 

3-6 

i.8W. 

O.2 

2.8 

5-3 

7-8 

10.3 

13.2 

16.0 

60 

55 

6.0 

4-6 

2.5 

0.4  W. 

2.O 

4.4 

7.0 

9-7 

12.3 

14.7 

55 

5O 

6.1 

4.8 

2.8 

0.7 

1.9 

4-3 

7.0 

9-6 

12.0 

14.0 

50 

45  N. 

6.0  W. 

4-7^. 

2.8  W. 

0.5  W. 

2.0  E. 

4-7E. 

7-3  E. 

9.8  E. 

H.8E. 

13-5  E. 

45  N. 

4O 

5-5 

4-4 

2.2 

o.o 

2-5 

5-o 

7.6 

10.0 

1  1.6 

13.0 

40 

35 

4.4 

3-4 

!-5 

0.5  E. 

3-i 

5-5 

7-8 

IO.O 

114 

12.4 

35 

3O 

2.9 

2.0 

0.4 

1.4 

3-8 

6.0 

8.1 

IO.O 

n.  i 

12.0 

30 

25 

1.6 

0.8 

0.5  E. 

2.2 

4-3 

6-4 

8.2 

IO.O 

II.O 

1  1.6 

25 

2O  N. 

0.5  W. 

0.5  E. 

1.5  E. 

3.1  E. 

5.0  E. 

6.8  E. 

8.4  E. 

9-7  E. 

10.5  E. 

10.8  E. 

2O  N. 

15 

0.5  E. 

1.2 

2-5 

3-9 

5-6 

7-i 

8.4 

9.4 

9.9 

IO.O 

15 

1O 

1.4 

2.2 

3-2 

4-7 

6.0 

74 

8.4 

9.2 

9-5 

9-5 

10 

5 

1.9 

2.9 

4.0 

5-2 

6.4 

7-7 

8.4 

9.0 

9-3 

9.0 

5 

O 

2-3 

3-4 

4.6 

5-7 

6.8 

8.0 

8.4 

8-7 

9.0 

8-7 

0 

]     5S. 

2.7  E. 

3.8  E. 

5.0  E. 

6.iE. 

7.2  E. 

8.2  E. 

8.6  E. 

9.0  E. 

9.1  E. 

8.8  E. 

58. 

1O 

3-o 

4.2 

5i 

6.6 

7.6 

8.4 

9.0 

9-3 

9-3 

9.2 

10 

15 

3-1 

4-5 

5.8 

7-i 

8.2 

9.0 

9-5 

9-7 

9-7 

9.6 

15 

2O 

3-2 

4.8 

6.3 

7-7 

8.8 

9-5 

fO.O         !  IO-3 

10.4 

IO.2 

20 

25 

3-3 

5-2 

7.0 

8-3 

9-5 

10.3 

ii.o      J  11.3 

11.4 

II.  I 

25 

22s- 

3-3  E. 

5.6  E. 

7.5  E. 

9.iE. 

10.3  E. 

11.3  E. 

12.0  E.    12.5  E. 

12.6  E. 

12.3  E. 

30  S. 

35 

3-2 

6.0 

8.1 

1  0.0 

"•3 

12.4 

13.2 

*3-7 

13-7 

l3-4 

35 

4O 

3>o 

6-3 

8.8 

10.7 

12.3 

13.6 

14.4 

14.9 

15.0 

14.6 

40 

45 

2.8 

6.6 

9-7 

11.7 

J3-5 

14.8 

15.7 

16.2 

16.3 

1  6.0 

45 

&O 

2.5 

7.0 

10.3 

12.7 

14.7 

16.2 

17.2 

17-7 

17.8 

17-5 

50 

5$  ft 

1.8  E. 

6.8  E. 

ii.oE. 

14.0  E. 

16.3  E. 

18.1  E. 

14.1  E. 

19.6  E. 

18.4  E. 

i9.oE. 

558. 

6O 

0.0 

6.0 

11.4 

15.6 

18.5 

20.7 

22.0 

22.4 

22.0 

21.2 

60 

174 


TABLE  LVII. 


Magnetic  Elements  of  the  Earth:    The  Magnetic  Dip. 

Longitude  West  of  Greenwich. 

i  Lat. 

Lat. 

o 
O 

10  |*0 

0  I         0 

3O    40 

si 

o        o 
60    70 

80 

90° 

10§ 

o 
110 

1*0 

130 

o 
140 

o          o  1        o  '        o 
150  160J170  ISO 

o 

o       o 

o  i      o 

o 

o  i         n 

o 

o 

r 

o 

o 

.  , 

75N. 

+8  1 

+82  +83 

+83+85 

+86  +86  +87 

+89 

+89 

+89 

+87 

+87 

+86 

+85° 

+83  :  +82 

+82  !  +82 

75N.J 

70 

79 

81 

81 

82!  84 

85 

85,  86 

88 

89 

89 

87 

86 

84 

83 

Si      80 

80  i    79 

7O 

65 

75 

76 

77 

78  i  79 

81    83!  85 

87 

87 

86 

85 

85 

83 

81 

79     77 

76     75 

65 

!60 

72 

73    74 

76    78 

80  i  82    84 

86 

86 

85 

83 

82 

79 

77 

75     73 

72!    7i 

60 

55 

70 

72    73 

74    76 

78    So:  82    83 

83 

82 

80 

78 

76 

74 

72     70 

68 

66 

55 

|50N. 

+67 

+69  +71 

+73  >75 

+77  j+78  +79  +80 

+79 

+78 

+76 

+74 

+72 

+69 

+57  +65 

+63  i  +6  1 

5ON. 

145 

64 

66    68 

7i     73 

75  I  76    77 

77 

76 

74 

72 

70 

68 

65 

63 

60 

58     56 

45 

J4O 

60 

62    65 

oa     70 

72      72;    72 

72 

71 

69 

67 

65 

62 

60 

57     55 

52 

51 

40 

35 

56 

58    61 

65    67 

68    63  ;  68 

67 

65 

64 

62 

59 

57 

55 

52     50 

48 

47 

35 

30 

54    57 

61    63    64!  64!  63    62 

60 

58 

56 

54 

52 

48 

47 

45      43 

30 

!  25  N. 

+46 

+5°  +54  i+57+58 

+59+58  +57+j6 

+53 

+51   +49 

+48 

+47 

+46 

+44   +43 

+42  i  +39 

25N. 

:  2O 

38 

43    48    52    54 

55  !  53    52 

49 

45 

43 

42 

42 

42 

41 

40 

38 

35  i   30 

2O 

1  5 

30 

37:  42 

47;  49 

49    47!  44 

40 

38 

36 

34 

33 

33 

32 

32 

3° 

27       22 

15 

10 

20 

29    30 

40    42    41  :  40  i  37 

33 

28     26 

24 

24 

23 

23 

21 

19 

14 

10 

5 

IO 

20      29 

33    35    34    33    3° 

27 

22 

17 

H 

13 

'3 

13 

13 

II 

+  9+5 

5 

0 

0  +I0j+20 

+26  :+28 

+27  +25  +20  +16  +10 

+  5 

+  3 

+   2 

+   2 

+   2 

+   2 

+    I 

-1-4 

0 

58. 

-  8+  2\    12 

19    21 

23      IS!    12  +   82+   2 

-3-5 

-  7 

-  7 

-  7 

—     7 

-  7 

10 

'4 

58. 

10 

18 

-6  +  4 

ii     14!  13      9  +  2-3-9 

12        14 

16 

16 

16 

16 

17 

20 

23 

10 

15 

28 

16-5 

+2+5+4      0-7 

12 

17 

20       22 

24 

24 

24 

26     27 

3° 

32 

15 

20 

34 

26  j  15 

-  8  -  4  -  5  -io;  15  i  20 

25 

28 

3° 

32 

33 

33 

34     35 

37 

39 

20 

25  S. 

-38 

32-24 

-i8!-i4  -14  -18  -23J-28  -32 

-35  -38 

-40 

-41 

-41  :  -41   -42 

-44 

-46 

258. 

30 

42 

37j  3i 

27    23!  23  :'  26!  30    34 

38 

44 

46 

47 

48  1    49 

50 

51 

53 

30 

35 

46 

41!  36 

33    3° 

3°    33'  36    4° 

44 

47 

5° 

51 

52 

53 

54     55 

56 

S8 

35 

!4O 

5° 

45    40 

38  i  36 

36    39  ,  42  ]  46 

50 

53 

55 

56 

57 

58 

59!   60 

61 

62 

40 

145 

53 

50    45 

43!  42 

42 

45j  48|  51 

55 

58 

60 

61 

62 

63 

63!   64 

65     66 

45 

508. 

-55 

-52  -49 

-47  j-47 

-48  -50  -52  -55 

-59 

-62 

-34 

-65 

-66 

-67  -67!  -68 

-69  -70 

5OS. 

55 

57 

55    53 

52!  52 

53    55l  57 

60 

64 

67 

68 

69 

70 

71     71 

72 

73 

73 

55 

6O 

59 

57 

56 

55    56 

57    59  1  62 

65 

68 

70 

72 

73 

74 

75 

75 

76 

76 

76 

60 

Longitude  East  of  Greenwich. 

Lat. 

Lat. 

c 
0 

10 

*o 

:n 

10 

50    60 

70° 

80° 

0 

100 

110 

1*0 

130 

145 

150 

160 

170 

180 

J75°N. 

+8? 

0           0 

+8  1  +8  1 

+80 

+sS 

O  j         O 

+8oj+8i 

+85 

+81 

+8? 

+81° 

+81 

+81 

+81 

+8? 

+8^ 

+8? 

~& 

+82° 

75N. 

7O 

79 

79 

78 

77 

77 

77|  78 

79 

79 

80 

80 

80 

80 

So 

80 

79 

•  79 

79 

79 

70 

65 

75 

75 

74 

74 

74 

75 

76 

76 

76 

77 

77 

77 

77 

77 

76 

76 

75 

74 

75 

65 

60 

72 

72 

71 

71 

71 

71 

72 

72 

72 

73 

73 

73 

73 

73 

72 

72 

70 

70 

60 

55 

70 

69 

68 

68 

68 

68 

69 

70 

70 

70 

69 

68 

66 

65 

66 

55 

50  N. 

+67 

+66 

+65 

+64 

+64 

+64 

+64 

+65 

+65 

fb5 

+66 

+66 

+66 

+65 

+64 

+63 

+6  1 

+60 

+5  1 

50  N. 

45 

64 

62 

61 

60 

60 

60 

60 

60 

60 

60 

60 

61 

61 

60 

59 

57 

55 

55 

56 

45 

40 

60 

57 

55 

54 

53 

53 

53 

53 

54 

55 

55 

56 

56 

55 

53 

49 

49 

5* 

40 

35 

56 

53 

5° 

48 

47 

47 

47 

48 

49 

50 

So 

50 

5° 

49 

48 

46 

44 

45 

47 

35 

3O 

51 

47 

43 

41 

40 

39 

39 

40 

40 

41 

42 

43 

43 

42 

42 

40 

40 

43 

30 

25  N. 

+46 

+40 

+35 

+31 

+30 

+29 

+29 

+30 

+31 

+33 

+34 

+35 

+35 

+35 

+35 

+33 

+33 

+35 

+39 

25N. 

2O 

38 

30 

23 

20 

19 

19 

20 

20 

21 

22 

23 

25 

27 

28 

28 

24 

24 

25 

30 

20 

15 

3° 

20 

14 

10 

10 

IO 

IO 

10 

IO 

II 

12 

15 

17 

18 

19 

15 

15 

22 

15 

10 

20 

10 

+  4 

0 

o 

0        0 

O 

0 

0 

+    I 

+  3 

+  5 

+  8 

9 

9 

8 

9 

14 

IO 

5 

IO 

0 

-  6 

-IO 

-10 

-10  -10 

-10 

-10 

-  9 

-  7 

-  5 

-  3 

-  i 

0 

o 

o 

o 

+  5 

5 

O 

O 

-  9 

-16 

-20 

-22 

-22 

-21 

-2O 

-20 

-18 

-16 

-14 

-12 

-n 

-IO 

-10 

-10 

-  9 

-  4 

0 

58. 

-  8 

19 

25 

30 

30 

3° 

30 

3° 

29 

27 

24 

22 

20 

20 

20 

21 

21 

19 

14 

58. 

IO 

18 

28 

33 

37 

39 

40 

40 

39 

38 

36 

33 

32 

31 

3° 

31 

32 

31 

28 

23 

10 

15 

27 

34 

40 

44 

47 

48 

48 

47 

46 

44 

43 

42 

41 

40 

40 

40 

38 

35 

32 

15 

20 

34 

40 

46 

50 

53 

54 

53 

52 

52 

51 

51 

51 

50 

49 

47 

46 

44 

42 

39 

2O 

25  S 
30 
35 

-38 
42 
46 

-44 
49 
52 

-5o 

-54 
57 

-57 
59 

63 

-57 

61 

64 

t 

64 

t 

65 

65 

-55 
59 
65 

-55 
59 
65 

-55 
59 
65 

1 

-53 

H 

-53 
57 
*3 

1 

-50 
§ 

-46 
53 

258. 
30 
35 

40 

5° 

54 

58 

61 

63 

65 

66 

67 

67 

68 

69 

70 

70 

69 

68 

67 

66 

64 

62 

40 

45 

53 

56 

60 

62 

65 

67 

68 

69 

70 

71 

72 

74 

75 

74 

73 

72 

70 

68 

66 

45 

508. 
55 
6O 

-55 
57 
59 

-f 
60 
62 

-6  1 
62 
64 

-63 

it 

-66 

11 

-68 

69 
70 

-69 
72 

72 

73 

-72 

74 
75 

-73 
76 
77 

1 

-76 

79 
82 

1o7 
83 

-£ 
«3 

-76 

79 
82 

18 

Si 

-74 

-72 
75 
77 

-70 

8 

508. 
55 

60 

TABLE  LVIII. 


175 


Magnetic  Elements  of  the  Earth  :  The  Horizontal  Force. 

Longitude  West  of  Greenwich. 

Lat. 

Lat. 

o 

0 

o 

10 

20 

o 
30 

40° 

50° 

60° 

o 
70 

80 

90 

o 
100 

HO  1*0 

13°0 

140 

150 

16°0 

o 
170 

o 
ISO 

75  N. 

0.6 

O.  ^ 

0.5 

0.4 

0.4 

0-3 

O.I 

0.0 

o.o 

o.o 

O.O 

o.o 

O.I 

0.2 

°-3 

0-3 

0.4 

0.4 

0.4 

75  N. 

70 

7 

6 

6 

5 

4 

3 

2 

o 

o 

o 

o 

0 

3 

4 

5 

5 

6 

6 

7 

70 

65 

8 

7 

7 

6 

5 

4 

3 

I 

o 

0 

I 

3 

4 

6 

6 

7 

8 

8 

9 

65 

60 

9 

8 

7 

6 

6 

4 

3 

3 

3 

4 

5 

6 

7 

8 

9 

I.O 

I.O 

I.O 

60 

55 

9 

9 

8 

8 

7 

6 

6 

5 

4 

5 

6 

7 

8 

9 

I.O 

I.O 

i 

2 

2 

55 

i50N. 

I.O 

I.O 

0.9 

0.9 

0.8 

0.7 

0.7 

0.6 

0.6 

0.7 

0.8 

0.9 

I.O 

i.i 

i.i 

1.2 

'•3 

1.3 

1.4 

50  N. 

145 

2 

i 

I.O 

9 

9 

8 

8 

8 

9 

I.O 

i.i 

i.i 

2 

2 

3 

4 

4 

4 

4 

45 

40 
35 

3 
4 

2 

3 

i 

2 

4" 

I.O 
2 

I.O 

i 

I.O 

i 

I.O 

2 

I.O 

2 

i.i 

3 

2 

4 
6 

3 
5 

3 

5 

J 

4 
6 

I 

I 

6 

6 

I 

6 

6 

40 

35 

25  N. 

1.6 

1-5 

1.4 

1.4 

1.6 

1.8 

1.9 

1.9 

1.9 

1.8 

1.8 

'•7 

25  N. 

2O 

7 

'6 

6 

5 

'e 

7 

8 

9 

2.O 

2.0 

2.O 

9 

9 

8 

7 

7 

7 

7 

20 

15 

8 

7 

7 

6 

6 

7 

8 

9 

2.O 

I 

I 

I 

2.0 

9 

8 

8 

8 

8 

8 

15 

10 

9 

8 

8 

7 

7 

8 

9 

2.0 

I 

2 

2 

I 

I 

2.O 

9 

889 

9 

10 

5 

9 

8 

8 

8 

8 

9 

2.0 

o 

I 

2 

2 

I 

I 

O 

9 

9 

9 

9 

9 

5 

0 

1.6 

1.9 

1.9 

1.9 

1.9 

1.9 

2.O 

2.0 

2.1 

2.2 

2.1 

2.1 

2.1 

2.0 

2.Q 

1.9 

1.9  i  2.0 

2.O 

O 

58. 

8 

8 

8 

9 

9 

0 

O 

I 

I 

I 

I 

0 

O         Q 

2.0 

2.O         O 

0 

58. 

10 

7 

7 

7 

8 

8 

9 

1.9 

0 

O 

I 

I 

O 

O 

0   i      G 

O 

O         O 

O 

10 

15 

6 

6 

6 

7 

7 

8 

'  8 

1.9 

O 

O 

O 

O 

O 

O         O 

O 

0    !      0 

0 

15 

2O 

4 

5 

6 

6 

6 

7 

8 

9 

1.9 

0 

0 

0 

0 

0         O         0 

oj    o 

O 

20 

258. 

14 

1.4 

T-5 

I-5 

1.6 

1.6 

J-7 

1.8 

1.9 

2.0 

2.O 

2.0 

1.9 

1.9 

1  9  j  1.9  i  1.9     1.9 

1.9 

258. 

3O 

3 

4 

4 

5 

5 

6 

7 

8 

9 

1.9 

1.9 

1.9 

9       8 

8       8 

7 

30 

35 

3 

3 

4 

5 

S 

6 

7 

8 

8 

8 

8 

0                    0 

7 

7 

7 

6 

35 

40 
45 

3 

3 

4 

4* 

5 

5 

6 
6 

6 

7 

8 

8 

8 

5 

I 

7 

6 

6 

6 

5 

5 

40 
45 

508. 

1.4 

5 

1.6 

1.6 

1.6 

7 
1.6 

1.6 

. 

1.4 

1.4 

1.4 

J-3 

J3 

I3 

1.2 

508. 

55 

3 

4 

4 

5 

5 

6 

6 

5 

5 

5 

4 

3 

3 

2 

2 

2 

i 

i 

0 

55 

60 

3 

4 

4 

5 

5 

5 

5 

5 

4 

3 

2 

i 

o 

0 

0.9 

0.9 

0.8 

0.8 

60 

Longitude  East  of  Greenwich. 

Lat. 

Lat. 

0 

»o 

20 

O            O 

30    40 

o 
50 

60 

o 
70 

80 

o 
90 

100 

110 

120 

130 

140 

o 

150 

o 

160 

o 

170 

o 
ISO 

75  N. 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

0.6 

o-S 

0-5 

0-5 

0.5 

0-5 

0-5 

0:5 

0.4 

75  N. 

70 

7 

7 

7 

8 

8 

8 

8 

8 

7 

7 

6 

5 

5 

6 

6 

6 

8 

8 

7 

70 

65 

8 

8 

8 

8 

8 

8 

8 

9 

9 

8 

8 

8 

8 

8 

8 

9 

9 

9 

9 

65 

6O 

9 

9 

9 

9 

9 

9 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

60 

55 

9 

I.O 

I.O 

i.i 

i.i 

i.i 

2 

2 

i 

i 

i 

i 

i 

i 

i 

2 

2 

2 

2 

55 

5ON 

I.O 

i.i 

T  T 

i  ? 

1.2 

i  ? 

I  3 

I  "\ 

i  "3 

i  ^ 

i  ^ 

1  1 

T  1 

1.2 

i  3 

I  "* 

1-4. 

5ON. 

45 

2 

2 

3 

3 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

T* 
4 

4 

45 

40 
35 

3 
4 

3 

4 

4 

5 

^ 

5 

7 

6 

7 

6 

7 

6 

7 

6 

7 

7 

6 

6 
7 

6 

7 

6 

i 

40 

35 

30 

5 

S 

6 

7 

8 

8 

9 

9 

2.0 

2.0 

9 

9 

8 

8 

7 

7 

6 

6 

6 

30 

25  N. 

1.6 

i-7 

1.7 

1.8 

1.9 

1.9 

2.0 

2.1 

2.2 

2.1 

2.0 

2.0 

1.9 

1.9 

1.8 

1.8 

i-7 

1.7 

1.7 

25  N. 

20 

7 

8 

9 

9 

9 

2.O 

0 

I 

3 

2 

I 

I 

2.O 

2.0 

9 

8 

8 

7 

7 

20 

15 

8 

9 

9 

9 

1.9 

O 

I 

2 

3 

2 

I 

I 

0 

2.O 

9 

9 

8 

8 

15 

1O 

9 

9 

8 

8 

9 

1.9 

0 

I 

2 

2 

2 

I 

I 

0 

2.O 

9 

9 

9 

10 

5 

9 

9 

8 

7 

7 

8 

9 

O 

I 

I 

2 

2 

2 

I 

I 

0 

2.0 

2.0 

9 

5 

O 

1.9 

1.8 

1.7 

1.6 

1.6 

1  7 

1.8 

1.9 

2.O 

2.1 

2.1 

2.1 

2.2 

2.2 

2.1 

2.1 

2.0 

2.0 

2.0 

O 

58. 

8 

7 

6 

5 

5 

6 

7 

8 

1-9 

0 

I 

I 

2 

2 

I 

I 

I 

0 

0 

58. 

10 

7 

6 

5 

5 

5 

5 

6 

7 

8 

1-9 

0 

I 

0 

I 

I 

I 

I 

I 

O 

1O 

15 

6 

- 

6 

7 

8 

T  rt 

j 

j 

I 

o 

Q 

o 

15 

20 

4 

4 

3 

3 

3 

3 

4 

5 

6 

7 

1 

1.8 

1.9 

0 

0 

0 

0 

o 

20 

258. 

1.4 

I>3 

I-3 

1'3 

x-3 

x-3 

*'3 

1.4 

I-5 

j  e 

1.6 

I-7 

1.8 

1.8 

1.8 

1.8 

1.8 

1.8 

1.9 

25  S. 

30 
35 

3 
3 

3 

2 

3 

2 

2 

2 

I 

2 

I 

2 

I 

3 

2 

4 

2 

4 

3 

5 

3 

6 

4 

6' 
4 

7 
4 

7 
4 

7 

5 

7 
5 

I 

I 

30 
35 

4O 

3 

2 

2 

I 

I 

I 

I 

I 

2 

2 

3 

3 

3 

3 

3 

4 

4 

5 

4O 

45 

3 

2 

I 

I 

0 

O 

O 

I 

I 

I 

i 

i 

2 

2 

2 

2 

3 

3 

45 

508. 

1-3 

1.2 

I.I 

I. 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

I.O 

i.o 

I.O 

I.O 

I.O 

I.O 

I.I 

i.i 

1.2 

508. 

55 

3 

2 

2 

0 

0 

0 

0.9 

0.9 

0.9 

0.8 

0.8 

0.7 

0.7 

0.8 

0.8 

0.9 

0.9 

0 

55 

6O 

3 

3 

2 

I 

o 

o 

9 

8 

8 

7 

6 

5 

5 

5 

5 

7 

8 

8 

6O 

176 


TABLE  LIX. 


Right  Ascension  of  the  True  Sun  and  Equation  of  Time. 

1875:    At  Greenwich  Mean  Noon. 

Da 

January. 

February. 

March. 

April. 

May. 

June. 

ay. 

0.R.A.     E.T. 

©,R.A.|   E.T. 

0.R.A.     E.T. 

0',  R.  A.     E.  T. 

0-.R.A.     E.T. 

0,11.  A.     E.T. 

h     m 

m 

1,       in                 m, 

h     in            in 

h     m            in 

h     'in            in 

//      m 

in 

1 

18  46.7 

-  3-8 

20   59.0 

-13-8 

22  48.1    —12.6 

o  41.8  —  4.0 

2  33-°  4-  3-° 

4  357 

2 

51-1 

4.2 

21     3-0 

14.0 

51.9        12.4 

454        3-7 

36-9        3-J 

39.8       2.4  : 

3 

55-5 

4-7 

7.1  1    14.1 

55.6         12.2 

49.0        3.4 

40.7 

3-2 

43-9 

2.2  j 

4 

59.9 

5-1 

II.  I 

14.2 

59-3 

I2.O 

52.7        3-i 

44-5 

3-3 

48.0 

2.0 

5 

19    4-3 

5-6 

15-2 

14-3 

23    3-o 

II.7 

56.3        2.8 

48.4 

34 

52-2 

•9 

6 

8-7 

6.0 

19.2 

H-3 

6.8 

II.5 

i     o.o        2.5 

52.2 

3-5 

56.3 

.7 

7 

I3-1 

6.5 

23.2 

14.4 

10.5 

"•3 

3-6  ,         2.2 

56.1 

3-6 

5    0.4 

.5 

^ 

19  17.4 

-6.9 

21    27.2 

—14.4 

23  14-2 

—  II.O 

i     7-3  —  2.9 

3    °-° 

+  3-6 

5    4-5 

+  .3 

9 

21.8 

7-3 

31.2 

14.5 

17.9 

10.8 

ii.o         1.7 

3-9 

3-7 

8.7 

.1 

1O 

26.2 

7-7 

35-1 

14-5 

21.5 

10.5 

14.6       1.4 

7-7 

3-8 

12.8 

0.9 

11 

3°-5 

8.2 

14-5 

25.2 

10.3 

18.3      I.I 

".7        3-8 

16.9 

0.7 

13 

39-2 

8.5 
8-9 

43-o 
46.9 

14.5 

14-5 

28.9 
32-6 

IO.O 

9-7 

22.O           0.9 

25.7        0.6 

15.6 
19-5 

!i 

21.  1 

25.2 

o-5 

14 

43-5 

9-3 

50.8 

14.4 

36-2 

94 

29-3  :     0.3 

234 

3-9 

294 

O.I 

15 

19  47.8 

—  9-7 

21  54-7 

—14.4 

23  39-9 

—  9.2 

i  33.0  —  o.i 

3  274 

-f  3-9 

5  33-5 

—   O.I 

16 

52.1 

IO.O 

58.6 

14-3 

43-5 

8.9 

36.7  -f-   0.2 

31.3 

3-8 

37-7 

0.3 

17 

56.3 

10.3 

22      2.5 

14-3 

47.2 

8.6 

40.4           0.4 

35-3 

3-8 

41.8 

°-5 

18 

20    0.6 

10.7 

6.4 

14.2 

50.8 

8.3 

44.1        0.6 

39-2 

3-8 

46.0 

0.7 

19 

4-9 

1  1.0 

IO.2 

14.1 

54-5 

8.0 

47.9        0.9 

43-2 

3-8 

50.1 

0.9 

20 

9.1 

"•3 

14.0 

14.0 

58.1 

7-7 

51.6          .1 

47.2 

3-7 

54-3 

1.2 

21 

M-5 

17.9 

13-9 

o     1.8 

74 

55-3  i        -3 

51.2 

3-7 

58.5 

1.4 

22 

20    17.5 

—  11.8 

22   21-7 

—13.8 

o    5.4!—  7.1 

i  59-o  4-    -5 

3  55-2 

-f  3-6 

6    2.6 

—  1.6 

23 

21.7 

I2.I 

25-5 

13.6 

9.0  1      6.8 

2    2.8         .7 

59-2 

3-5 

6.8 

1.8 

24 

25.9 

12.3 

29-3 

13-5 

12.7 

6.5 

6.5         .9 

4    3-2 

3-5 

10.9 

2.0 

25 

3O.I 

12.6 

33-i 

13-3 

16.3 

6.2 

10.3           2.1 

7-3 

34 

15.1 

2.2 

26 

34-3 

12.8 

36.8 

19.9 

5-8 

I4.I  !         2.2 

"•3 

3-3 

19.2 

2.4 

27 

13.0 

40.6 

13.0 

23-6 

5-5 

I7.8           2.4 

154 

3-2 

234 

2.7 

28 

42.6 

13.2 

444 

12.8 

27.2 

5-2 

21.6           2.6 

19.4 

3-° 

27.6 

2.9 

29 
30 

20  46.7 
50.8 

54-9 

-134 
13.5 

o  30.8 

1! 

-4-9 
4.6 

4-3 

2  25.4  4-  2.7 

29.2           2.9 

4  23.5  4-  2.9 
27.6        2.8 

31.6    2.7 

6  31.7 
35.9 

3-3 

Da 

July. 

August. 

September. 

October. 

XoTember. 

December. 

ay. 

0'.R.A. 

E.  T. 

0.R.A. 

E.T. 

0'SR.A. 

E.T. 

©'.R.A. 

E.T. 

08R.A. 

E.T. 

0',R.A. 

E.T. 

h     m 

m 

h     m 

m 

h    m 

in 

h    m 

m 

h     m 

m 

h     m 

m 

1 

6  40.0 

—  3-5 

844-8 

—  6.1 

10  40.9 

4-   0.0 

12  29.0 

+  10-3 

14  25.2 

+16.3 

1  6  28.9 

2 

44.1 

3-6 

48.7 

6.0 

44-5 

0.4 

32.6 

10.6 

29.1 

16,3 

33-2 

10.5 

3 

48.2 

3-8 

52.6 

6.0 

48.2 

0.7 

36.2 

10.9 

33-0 

I6.3 

37-5 

IO.I 

4 

52.4 

4.0 

56.5 

CQ 

51.8 

1.0 

39-9 

1  1.2 

37-0 

16.3 

41.9 

9-7 

5 

56.5 

4.2 

9    °-3 

5.8 

554 

i*3 

43-5 

II-5 

40.9 

16.3 

46.2 

9-3 

6 

7    0.6 

44 

4.2 

5-7 

59-o 

47-2 

u.8 

44.9 

16,3 

50.6 

8.9 

7 

4.7 

4-5 

8.0 

5.6 

II      2.6 

2.0 

50.8        I2.I 

48.9 

16.2 

55-0 

8.4 

8 

7    8.8 

—  4-7 

9  u.8 

—  54 

II      6.2 

+  2.4 

12    54.4  4-12.4 

14  52.9 

4-16.1 

16  59-3 

4-  8.0, 

9 

12.9 

4-8 

15.6 

5-3 

9.8 

2.7 

58.2 

12.6 

56-9 

16.1 

i7    3-7 

7.6 

10 

17.0 

5.0 

19.4 

5-2 

134 

3'° 

13     1.8 

12.9 

15     i.o 

16.0 

8.1 

7-1 

11 

21.  1 

S-2 

23-2 

5-° 

17.0 

34 

5-5 

13.2 

5-° 

15-9 

12.5 

6.6 

12 

25.2 

5-3 

27.0 

4-9 

2O.6 

3-7 

9.2 

134 

9.1 

15-7 

16.9 

6.2 

13 

29.2 

54 

30.8 

4-7 

24.2 

4.1 

12.9 

13.7 

13.2 

l$.6 

21.4 

5-7 

14 

33-3 

5-5 

34-5 

4-5 

27.8 

44 

1  6.6 

13.9 

17.2 

15-5 

25.8 

5-2 

15 

7  37-3 

-5-6 

9  38-3 

—  4-3 

II  31.4 

+  4-8 

13  20.3 

4-14.1 

15  21.4 

+15,3 

17  30.2 

+  4-7 

16 

41.4 

5-7 

42.0 

4.1 

35-o 

5-1 

24.0 

14.4 

25-5 

I5<1 

34-6 

4-3 

17 

454 

5-8 

45-7 

3-9 

38.6 

5-5 

27.8 

'4-5 

29.6 

14.9 

3-* 

18 

49-5 

5-9 

49-5 

3-7 

42.2 

5-8 

3L5 

14.8 

33-8 

14.7 

43-5 

3-3' 

19 

53-5 

6.0 

53-2 

3-5 

45-7 

6.2 

35-3 

14.9 

37-9 

14.5 

47-9 

2.8 

20 

57-5 

6.0 

56-9 

3-3 

49-3 

6-5 

39-0 

I5*1 

42.1 

14-3 

524 

2.3! 

21 

8    0.5 

6.1 

10    0.6 

3-0 

52-9 

6.9 

42.8 

15-3 

46.3 

14.0 

56.8 

1.8 

22 

8    5-5 

—   6.2 

10    4.3 

—  2.8 

ii  56.5 

+  7-2 

13  46.6 

+  154 

15  50-5 

+13-8 

18     1.2 

4-  1.3 

23 

9-5 

6.2 

8.0 

2-5 

12      O.I 

7.6 

504 

15.6 

54-7 

i3-5 

5-7 

0.8 

24 
25 

134 
17.4 

6.2 
6.2 

11.7 
15.4 

2-3 
2.0 

3-7 
7-3 

7-9 
8-3 

$ 

i5-7 

15.8 

58-9 
16    3.2 

13.2 
12.9 

IO.I 

14.6 

—  d* 

26 

21.3 

6.2 

19.0 

1.7 

10.9 

8.6 

14  1.9 

15-9 

74 

12.6 

19.0 

0.7! 

27 

25-3 

6.2 

22.7 

1-4 

14-5 

9.0 

5-7 

1  6.0 

»•  7 

12.3 

23-5 

1.2  | 

28 

29.2 

6.2 

26.4 

1.2 

18.1 

9-3 

9.6 

16.1 

16.0 

11.9 

27.9 

1.7 

29 

8  33-i 

—  6.2 

10  30.0 

—  0.9 

12   21-7 

4-  9.6 

14  "34 

4-16.2 

1  6  20.3 

4-11.6 

18  32-3 

—   2.2 

30 

37-1 

6.2 

33.6 

0.5 

254 

9-9 

17.4 

16.2 

24.6 

1  1.2 

36.8 

2-7 

31 

40.9 

6.1 

37-3 

0.3 

21.2 

16.3 

41.2 

3-2 

TABLE  LlX. 


177 


ISig'Sat  Ascension  of  the  True  §1111  and  Equation  of  Time. 

1876:    At  Greenwich  Mean  Noon. 

Hiiv 

January. 

February. 

March. 

April. 

M.iy. 

June. 

uaj. 

©'SR,A.     E.  T. 

0'sR.A.    E.T. 

0'SR.A. 

E.  T. 

©•SR.A. 

E.T. 

0'SR.  A. 

E.T. 

0SR.A.     E.T. 

h     'in    |        tn 

h    m 

tn 

h     in 

m 

h     m 

in, 

7l       Trt 

m 

h     m            in 

1 

18  45.6 

-3-6 

20  58.0 

-I3.8 

22  51.0 

—12.5 

0  44-5 

-3-8 

2  35-9 

+  3-1 

4  38.8  +  2.4 

2 

50.0 

4.1 

21      2.1 

13-9 

54-7 

12.2 

48.2 

3-5 

39-8 

3-2 

43-0           2.2 

3 

54-4 

4.6 

6.1 

14.0 

58.4 

12.6 

51.8 

3-2 

43-6 

3-3 

47-1            2.1 

4 

58.8  i       5.0 

10.2 

14.2 

23      2.2 

11.8 

55-5 

2.9 

474 

34 

51.2            1.9  i 

5 

19    3-2       5-5 

14.2 

14.2 

5-9 

1  1.6 

59-1 

2.6 

51-3 

3-5 

55-3        i-7 

6 

7.6 

5-9 

18.2 

14-3 

9.6 

M-3 

I      2.8 

2-3 

55-i 

3-6 

594 

i-5 

7 

12.0 

6.4 

22.2 

14.4 

13-3 

11.  i 

6.4 

2.0 

59-° 

3-6 

5    3-5 

1.4 

8 

19    16.4 

_  6.8 

21    26.2 

—14.4 

23  17-° 

—10.8 

I    IO.I 

—    1-7 

3    2.9 

+  3-7 

5     7-7 

+  1.2! 

9 

20.7 

7.2 

30.2 

H-5 

20.7 

10.6 

13-7 

1.4 

6.8 

3-8 

11.8 

I.O 

10 

25.1 

7.6 

34-2 

14-5 

24.3 

10.3 

17.4 

1.2 

10.7 

3-8 

15-9 

0.8 

11 

29.4 

8.0 

38.1 

14-5 

28.0 

IO.I 

21.  1 

0.9 

14.6 

3-8 

20.  i 

0.6 

12 

33-8 

8.4 

42.0 

H-5 

3I-7 

9.8 

24.8 

0.7 

18.5 

3-9 

24.2 

0.41 

13 

38.1 

8.8 

46.0 

14-5 

35-3 

9-5 

28.5 

0.4 

22.5 

3-9 

28.4 

0.2 

14 

42.4 

9.2 

49.9 

14.4 

39-o 

9.2 

32.1 

O.I 

26.4 

3-9 

32.5 

o.o 

15 

19  46.7 

—  9-5 

21  53-8 

—14.4 

23  42.6 

-  8.9 

i  35-8 

+  o.i 

3  3°4 

H-3-9 

5  36-7 

—  0.2 

16 

51.0 

9-9 

57-7 

H-3 

46-3 

8.6 

39-5 

o-3 

34.3 

3-8 

40.8 

0.4 

17 

55-3 

IO.2 

22      1-5 

H-3 

49-9 

8-3 

43-2 

0.6 

38.3 

3-8 

45-o 

0.7 

IS 

59-6 

10.6 

5-4 

14.2 

53-6 

8.1 

47.0 

0.8 

42.3 

3-8 

49.2 

0.9 

19 

20    3.8 

10.9 

9-3 

14.1 

57-2 

7-7 

5°-7 

I.O 

46.2 

3-8 

53-3 

20 

8.1 

II.  2 

14.0 

o    0.9 

7-5 

54-4        1.2 

50.2 

3-7 

57-5 

*-3 

21 

12.3 

"•5 

16.9 

13-9 

4-5 

7-1 

58.1 

i-5 

54-3 

3-6 

6     1.6 

i-5 

22 

20   16.5 

—11.7 

22    20.8 

-13.8 

o    8.1 

—  6.8 

2     1.9  +  1.6 

3  58.3 

+  3-5 

6    5.8  j—  1.8; 

23 

20.7 

I2.O 

24.6 

13-7 

11.8 

6.5 

5.6  j       1.8 

4    2.3 

3-5 

10.0 

2.O 

24 

24.9 

12-3 

28.4 

13-5 

iS-4 

6.2 

9.4 

2.O 

6-3 

34 

14.1 

2.2 

25 

29.1 

12-5 

32.2 

13-4 

19.1 

5-9 

13.2 

2.2 

10.3 

3-3 

18.3 

2.4 

26 

33-3 

12.7 

35-9 

13.2 

22.7 

5-6 

16.9 

2.4 

14.4 

3-2 

22.4 

2.6 

27 

374 

12.9 

39-7 

13.0 

26.3 

5-3 

20.7 

2-5 

18.5 

3-o 

26.6 

2.8 

28 

41.6 

I3.2 

43-5 

12.8 

30.0 

5-o 

24-5 

2.7 

22.5 

2.9 

30-7 

3-o 

29 

20   45.7 

—  !3-3 

22   47.2 

-12.6 

o  33-6 

—  4-7 

2   28.3 

+   2.8 

4  26.6 

+   2.8 

6  34-9 

—  3-2  i 

30 

49.8 

13-5 

37-3 

4.4 

32.1 

3-o 

3°-7 

2-7 

39-0  1      34 

31 

53-9 

13-7 

40.9 

4.1 

34-8 

2-5 

I  Day 

1 

July.                   August. 

September. 

October. 

\ovember. 

December. 

0'3R.A. 

E.  T.   S©?,R,A.     E.  T. 

0'SR.A.     E.T. 

0's  R.  A. 

E.T. 

0'sR.A.     E.T. 

0'sR.A.     E.T.  i 

j 

h     m 

m 

h     tn            in 

h     m 

m, 

h     m 

tn 

h     m 

m 

h     m    1        m    \ 

1 

6  43.1 

-3-6 

8  47.8 

—  6.0 

10  43-7 

+  0.3 

12   31-7 

+  10.5 

14  28.1 

+  16.3 

16  32.2 

+10.6 

2 

47-3 

3-8 

5J-7 

6.0 

47-3 

0.6 

354 

10.8 

32.1 

I6.3 

36.5 

10.2 

3 

5i-4 

4.0 

55-5 

5-9 

50-9 

0.9 

39-o 

n.  i 

36.0 

I6.3 

40.8 

9.8 

4 

55-5 

4.2 

59-4 

5-8 

54-5 

M 

42.6 

11.4 

40.0 

I6.3 

45-2 

94 

5 

59-6  !      4-3 

9    3-2 

5-7 

58.1 

1.6 

46.3 

11.7 

43-9 

I6.3 

49-5 

O.O 

6 

7    3-7 

4-5 

7.0 

5.6 

H     1.7 

1.9 

49-9 

12.0 

47-9 

16.2 

53-9 

8.5 

7 

7.8 

4.6 

10.9 

5-5 

5-3 

2-3 

53-6 

I2.3 

5i-9 

16.2 

58.3 

8.1 

8 

7  1  1-9 

-4-8 

9  H-7 

—  5-3 

ii    9.0 

+  2.6 

12  57.2 

+  12.6 

IA    56.0 

+  16.1 

17    2.7 

-f  7-7 

9 

16.0 

5-o 

18.5 

S-2 

12.5 

3-o 

13    0.9 

12.9 

15    o.o 

16.0 

7-i 

7.2 

1O 

20.1 

5-i 

22.3 

5-i 

16.1 

3-3 

4.6 

13.1 

4.0 

15-9 

"•5 

6.7 

11 

24.2 

S-2 

26.1 

4.9 

19.7 

3-7 

8-3 

134 

8.1 

15.8 

15-9 

6.3 

12 

28.3 

5-4 

29.8 

4-7 

23-3 

4.0 

I2.O 

13.6 

12.2 

15.6 

20.3 

5.8 

13 

32.3 

5-5 

33-6 

4-5 

26.9 

4.4 

15-7 

13-9 

I6.3 

15.5 

24.7 

5-3 

14 

36.4 

5-6 

37-4 

'    4-4 

3^-5 

4-7 

19.4 

14.1 

20.4 

15-3 

29.1 

4-9 

15 

7  40-4 

—  5-7 

9  4i-i 

—  4.2 

H    34.1 

+  -5-I 

13   23.1 

+14.3 

15    24-5 

+15-2 

17  33-6 

-h  44  ' 

16 

44-5 

5-8 

44-8 

3-9 

37-7 

5-4 

26.9 

14-5 

28.6 

15.0 

38.0 

3-9 

17 

48.5 

5-9 

48.6 

3-8 

4i-3 

5.8 

3O.6 

14.7 

32.8 

14.8 

42.4 

34 

18 

52.5 

6.0 

52.3 

3-5 

44.9 

6.1 

344 

14.9 

36.9 

14.6 

46.9 

2.9 

19 

56.5 

6.0 

56.0 

3-3 

48.5 

6.4 

38.1 

JS-1 

4I.I 

14-3 

5'-3 

2.4 

20 

8    0.5 

6.1 

59-7 

3-1 

52-1 

6.8 

41.9 

15.2 

45-3 

14.1 

55-8 

1.9 

21 

4-5 

6.1 

10      3.4 

2.8 

55-7 

7-i 

45-7 

154 

49-5 

13.8 

18    0.2 

1.4 

22 

8    8.5 

—  6.2 

10     7.1 

—   2.6 

n  59-3 

+  7-5 

13  49-5 

-H5-5 

15  53-7 

+13-6 

1  8    4.6 

+  0.9 

23 

12.5 

6.2 

10.8 

2-3 

12      2.9 

53-3 

i5-7 

57-9 

13-3 

9.1 

0.4 

24 

16.4 

6.2 

14-5 

2.1 

6.4 

8.2 

57-i 

15.8 

l6      2.2 

13.0 

*3-5 

—  o.i 

25 

20.4 

6.2 

18.2 

1.8 

10.  1 

8-5 

14     i.o 

i5-9 

6.4 

12.7 

18.0 

0.6 

26 

24-3 

6.2 

21.8 

i-5 

I3.6 

8.9 

4.8- 

16.0 

10.7 

12.3 

22.4 

i.i 

27 

28.2 

6.2 

25-5 

1.2 

17-3 

9.2 

8-7 

16.1 

15.0 

I2.O 

26.8 

1.6 

28 

32-2 

6.2 

29.1 

0.9 

20-9 

9-5 

12.5 

16.2 

19.2 

II-7 

31-3 

2.1 

29 

8  36.1 

—   6.2 

10  32.8 

—  0.6 

12   24.5 

+  9-9 

14  16.4 

+16.2 

16  23.5 

+  II.3 

18  35-7 

-   2.6 

30 

40.0 

6.1 

36-4 

0.3 

28.1 

10.2 

20.3 

16.3 

27.8 

II.O 

40.1 

3-i 

31 

43-9 

6.1 

40.0 

o.o 

24.2 

16.3 

44-5  !      3-5  ! 

178 


TABLE  LIX. 


Right  Ascension  of  the  True  Sun  and  Equation  of  Time. 

1877:    At  Greenwich  Mean  Noon. 

Dav 

January. 

February. 

March. 

April. 

May. 

June. 

ItU)  . 

©'>B.A. 

E.T. 

0'.B.A. 

E.T. 

0',B.A. 

E.T. 

0's  B.  A. 

E.T. 

©'SB.A. 

E.T. 

0'sR.A. 

E.T. 

h     m 

m 

h     in 

at 

h     m 

m 

h     m 

m 

h     in 

m 

h     in 

in   \ 

I 

1  8  49.0 

—  4.0 

21      I.I 

—13-9 

22  50.1 

—12.5 

o  43-7 

~  3-9 

2  35-o 

4-  3-1 

4  37-9 

+  24  i 

2 

534 

4-5 

5-i 

14.0 

53-8 

.12-3 

47-3 

3-6 

38.8 

3-2 

4i-9 

2-3 

3 

57.8 

4.9 

9-2 

14.1 

57-5 

I2.I 

5°-9 

3-2 

42.7 

3-3 

46.1 

2.1 

4 

19      2.2 

54 

13.2 

14.2 

23    i-3 

1  1.0 

54-6 

3-° 

46.5 

34 

50.2 

1.0 

5 

6.5 

5.8 

17-3 

14-3 

C.O 
3 

n.6 

58.2 

2.7 

5°4 

3-5 

54-3 

1.8 

6 

10-9 

6-3 

21.3 

14.4 

8.7 

11.4 

i     1.9 

2.4 

54-2 

3-6 

584 

i.6l 

7 

15-3 

6.7 

25-3 

14.4 

12.4 

II.  2 

5-5 

2.1 

58.1 

3-6 

5    2.5 

M- 

8 

19    19.7 

—  7.1 

21    29.3 

-14-5 

23  16.1 

—  10-9 

i    9.2 

—  1.8 

3    2.0 

+  3-7 

5    6.7 

+    1.2 

9 

24.0 

7-5 

33-2 

14-5 

19.8 

10.7 

12.9 

i-5 

5-9 

3-7 

10.8 

I.O 

10 

28.4 

7-9 

37-2 

14-5 

23-5 

IO.4 

16.5 

1.2 

9.8 

3-8 

14.9 

o.S 

11 

32.7 

8-3 

41.1 

14-5 

27.1 

IO.I 

20.2 

I.O 

13-7 

3-8 

19.1 

0.6; 

12 

37-i 

8.7 

45  '° 

14-5 

30.8 

9.9 

23-9 

0-7' 

17.6 

3-8 

23.2 

0.4 

13 

41.4 

9.1 

49.0 

14-5 

34-5 

9.6 

27.6 

o-5 

21.5 

3-9 

27.4 

0.2 

14 

45-7 

9-5 

52-9 

14.4 

38-1 

9-3 

31-3 

0.2 

254 

3-9 

3i'S 

0.0 

15 

19  50.0 

-9.8 

21    56.8 

—14.4 

23  4i-8 

—  9.0 

i  35-o 

-f-   O.O 

3  294 

+  3-9 

5  35-7 

—   O.2 

16 

54-3 

10.2 

22      0.6 

14-3 

454 

8-7 

38.6 

0.3 

334 

3-8 

39-8 

0.4 

17 

58.6 

10.5 

4-5 

14.2 

49.1 

8.4 

42.4 

o-5 

37-3 

3-8 

44.0 

0.6! 

18 

20     2.8 

10.8 

8.4 

14.2 

52-7 

8.1 

46.1 

0.7 

41-3 

3-8 

48.1 

0.8 

19 

7.0 

ii.  i 

12.2 

14.1 

564 

7.8 

49.8 

I.O 

45-3 

37 

52-3 

I.O 

20 

n-3 

11.4 

16.1 

14.0 

o    o.o 

7-5 

53-5 

1.2 

49-3 

3-7 

56.5 

1-3 

21 

15-5 

11.7 

19.9 

13.8 

3-7 

7-3 

57-3 

1.4 

53-3 

3-6 

6    0.6 

22 

20    19.7 

—  12.0 

22   23.7 

—13-7 

o    7-3 

-6.9 

2      I.O 

4-  1.6 

3  57-3 

+  3-6 

6    4.8 

—  1-7; 

23 

23-9 

12.2 

27-5 

13-6 

10.9 

6.6 

4-7 

1.8 

4     i-3 

3-5 

8.9 

24 

28.1 

12.5 

31-3 

134 

14.6 

6-3 

8.5 

2.0 

54 

34 

13.1 

2.1 

25 

32-3 

12.7 

35-° 

13.2 

18.2 

6.0 

12.3 

2.1 

94 

3-3 

17.2 

2-3  : 

26 

364 

12.9 

38.8 

I3-I 

21.8 

5-7 

16.0 

2-3 

134 

3-2 

21.4 

2.6 

27 

40.6 

I3-I 

42.6 

12.9 

25-5 

54 

19.8 

2-5 

17-5 

3.1 

25.6 

2.8- 

28 

44-7 

13-3 

46.3 

12.7 

29.1 

23.6 

2.6 

21.5 

3-o 

29.7 

3-° 

29 

20  48.8 

—13-5 

o  32.7 

-4-8 

2   27.4 

+    2.8 

4  25.6 

4-  2.9 

6  33-9 

—  3-2 

30 

52-9 

13.6 

364 

4-5 

31.2 

2.9 

29.7 

2.7 

38.0 

34 

31 

57-o 

I3.8 

40.0 

4.2 

33-8 

2.6 

Dav 

July. 

August. 

September. 

October. 

November. 

December. 

u»y. 

0'8B.A. 

E.T. 

0'sR.A. 

E.T. 

0'sB.A. 

E.T. 

0'SB.A. 

E.T. 

0'.,R.A. 

E.T. 

0'.,B.A. 

E.T. 

h     m 

m 

h     m 

m 

h     m 

m 

h     m 

in 

h     m 

in 

h     m 

m 

1 

6  42.1 

—  3-5 

8-  46.8 

—  6.0 

10  42.8 

+   0.2 

12  30.8 

+  10-5 

14  27.2 

4-16-3 

16  31.1 

+  10.7: 

2 

46.2 

3-7 

50-7 

6.0 

46.4 

°-5 

34-5 

10.7 

3I-I 

I6.3 

354 

10.3! 

3 

5°4 

3-9 

54-6 

C.Q 

50.0 

0.9 

38.1 

II.  I 

35-i 

I6.3 

39-8 

9-9: 

4 

54-5 

4-i 

584 

5  • 

53-7 

1.2 

41.7 

11.4 

39-o 

I6.3 

44.1 

9-5! 

5 

58.6 

4-3 

9    2.3 

5-7 

57-3 

i-5 

454 

II.7 

43-o 

I6.3 

48.5 

Q«  I 

6 

7    2.7 

44 

6.1 

5-6 

ii    0.9 

1.8 

49.0 

12.0 

47.0 

16.2 

52.9 

o.o 

7 

6.8 

4.6 

9-9 

5-5 

4-5 

2.2 

52.7 

12.2 

51.0 

16.2 

57-2 

8.2 

8 

7  10.9 

-4.8 

9  13-8 

—  54 

ii    8.1 

+   2-5 

12   56.4 

+  12-5 

H  55-o 

+16.1 

17     1.6 

4-  7.8! 

9 

15.0 

4.9 

17.6 

S-2 

11.7 

2.9 

13    o-o 

12.8 

59-° 

16.0 

6.0 

7-3 

1O 

19.1 

21.4 

I5*3 

3.2 

3-7 

13.0 

15    3-i 

15-9 

10.4 

6.9 

11 

23.2 

5-2 

25.1 

4-9 

18.9 

3.6 

74 

13-3 

15.8 

14.8 

6.4 

12 

27-3 

5-3 

28.9 

4.8 

22.5 

3-9 

ii.  i 

13-5 

1  1.  2 

15-7 

19.2 

5-9 

13 

54 

32-7 

4.6 

26.1 

4.2 

14-8  1     13-8 

15-3 

15-5 

23.6 

5-5 

14 

354 

5.6 

36.5 

44 

29.7 

4.6 

18.5      14.0 

194 

154 

28.1 

5.0 

15 

7  39-5 

—  5.7 

9  40.2 

-  4.2 

ii  33-2 

4-  5.0 

13  22.3  +14.2 

15    23-5 

4-15.2 

17  32.5 

+  4-5 

16 

43-5 

5-8 

43-9 

4.0 

36.8 

5-3 

26.0 

14.4 

27.6 

15.0 

36-9 

4.0 

17 

47-5 

5-9 

47-7 

3-8 

40.4 

29.7 

14.7 

31.8 

14.8 

41.4 

3-5 

18 

5-9 

3-6 

44-o 

6.0 

33-5 

14.8 

35-9 

14.6 

45-8 

3-° 

19 

55-6 

6.0 

55  i 

.  34 

47.6 

6.4 

37-2 

15-0 

40.1 

14.4 

50.2 

2-5 

20 

59.6 

6.1 

58.8 

51.2 

6.7 

41.0 

15-2 

44-3 

14.1 

54-7 

2.0 

21 

8    3-5 

6.1 

10    2.5 

2.9 

54-8 

44-8 

15-3 

48.5 

13-9 

59-1 

i-5 

22 

8    7-5 

—    6.2 

10      6.2 

-  2.6 

n  584 

+  74 

13  48.6 

-H5-5 

15  52-7 

+  13.6 

18    3-5 

+    I.I 

23 

6.2 

9.9 

2.4 

12      2.O 

7.8 

524 

15.6 

55-9 

134 

8.0 

o-5 

24 

jCC 

6.2 

13.6 

2.1 

5-6 

8.1 

56.2 

15.8 

16     i.i 

13*1 

12.4 

O.I 

25 

19.4 

6.2 

17.2 

1.8 

9.2 

8.5 

14    o.o 

i5-9 

54 

12.8 

16.9 

-  0.5 

i  26 

234 

6.2 

20.9 

1.6 

12.8 

8.8 

3-9 

ib.o 

9.6 

12.5 

21.3 

I.O 

27 

28 

27-3 
31.2 

6.2 
6.2 

24.6 

28.2 

I.O 

16.4 

20.0 

9.1 
9-5 

16.1 
16.1 

13-9 
18.2 

I2.I 

u.8 

25-7 
30.2 

14 

1-9 

29 

—  6.2 

10  31.9 

—  0.7 

12  23.6 

+  9.8 

14  15-5 

+16.2 

16  22.5 

+"4 

18  34.6 

-  2.4 

30 

39-° 

6.1 

35-5 

0.4 

27.2 

IO.I 

19.4 

1  6.2 

26.8 

ii.  i 

39-o 

2.Q 

31 

42.9 

6.1 

39-2 

O.I 

23-3 

16.3 

43-5 

34 

TABLE  LIX. 


179 


Right  Ascension  of  tlae  True  Sun  and  Equation  of  Time. 

1878  :   At  Greenwich  Mean  Noon. 

j 

AAV 

January. 

February. 

March. 

April. 

May. 

June. 

VRJi 

0'SR.A. 

E.  T. 

©sB.A. 

E.T. 

0'.,B.A. 

E.T. 

0'sB.A. 

E.T. 

0'sB.A. 

E.T. 

0SB.A. 

E.T. 

h      in 

771 

h      m 

in 

k      m 

m 

h     in 

m 

h     m 

m 

h     m 

in 

1 

18  47.9 

—  3-9 

21      O.I 

-.-13.9 

22   49-2 

—12.5 

o  42.8 

-  3-9 

2  34-1 

+  3-0 

4  36.9 

+  2.4 

2 

52-3 

4-3 

4.2 

14.0 

52-9 

12.3 

46.4 

3-6 

37-9 

3-i' 

41.0 

2-3! 

3 

56.7 

4.8 

8.2 

14.1 

56.6 

la.i 

50.0 

3-3 

41.8 

3-2 

45-1 

2.1  i 

4 

19     i.i 

5-3 

12.3 

14.2 

23      04 

11.9 

53-7 

3-° 

45-6 

3-3 

49-2 

2.o! 

9 

5-5 

5-7 

I6.3 

14-3 

4.1 

11.7 

57-3 

2-7 

494 

34 

53-3 

1.8! 

6 

9-9 

6.2 

20.3 

14.4 

7-8 

M-5 

I       I.O 

2-5 

53-3 

3-5 

574 

1.6 

* 

14-3 

6.6 

24-3 

14.4 

"•5 

II.  2 

4-7 

2.2 

57-2 

5     1.6 

1.4 

8 

19  18.6 

-  7-o 

21    28.3 

-14-5 

23  15.2 

—  II.O 

i     8.3 

—  i-9 

3     i.o 

+  3-7 

5    5-7 

+    1.2 

9 

23.0        7.4 

32.3 

14.5 

I8..9 

10.7 

12.0 

1.6 

4.9 

3-7 

9-8 

I.I 

10 

27-3 

7-8 

36.2 

14-5 

22.6 

10-5 

I5.6 

i-3 

8.8 

3-8 

14.0 

0.8 

11 

31-7 

8.2 

40.2 

H-5 

26.2 

IO.2 

19-3 

I.O 

12.7 

3-8 

18.1 

0.7 

12 

36.0 

8.6 

44.1 

H-5 

29.9 

9-9 

23.0 

0.8 

16.7 

3-8 

22.2           0.5 

13 

40.3 

9.0 

48.0 

14.5 

33-6 

9-7 

26.7 

o-5 

2O.6 

3-8 

26.4  !         0.3 

14 

44.6 

94 

51-9 

14.4 

37-2 

9-3 

3°4 

o-3 

24.5 

3-9 

30.5  !       o.i 

15 

19  48.9 

—  9-7 

21    55.8 

—14.4 

23  40.9 

—  9.1 

i  34-i 

o.o 

3  28.5 

+  3-8 

5  34-7  —  0.2 

16 

53.2       10.1 

59-7 

14-3 

44.6 

8.8 

37-8 

4-   0.2 

324 

3-8 

38.9  !      0.4 

17 

57-5  !     i°4 

22    3.5 

14.2 

48.2 

8.5 

41.4 

0-5 

364 

3-8 

43-° 

0.6 

18 

20     i.8j     10.7 

74 

14.2 

51.8 

8.2 

45-2 

0.7 

40.4 

3-8 

47.2 

0.8  i 

19 

6.0 

II.O 

"•3 

14.1 

55-5 

7-9 

48.9 

0.9 

44-3 

3-8 

5!-3 

I.O 

2O 

10.2 

n-3 

15.1 

14.0 

59-i 

7.6 

52.6 

I.I 

48-3 

3-7 

55-5 

1.2 

21 

H-5 

11.6 

18.9 

I3.8 

0      2.8 

7-3 

56.3 

1.4 

52.3 

3-6 

59-6 

1-4 

22 

20    18.7 

—11.9 

22   22.7 

—13-7 

o    6.4 

—  7-o 

2      O.I 

+  1.6 

3  56-3 

+  3-6 

6    3.8 

—  i-7i 

23 

22.9 

12.  1 

26.5 

13-6 

10.0 

6.7 

3-8 

1.8 

4    0.4 

3-5 

8.0 

1.9! 

24 

27.1 

12.4 

30-3 

134 

13-7 

6.4 

7.6 

1.9 

44 

34 

I2.I 

2.1 

J25 

31.2 

12.6 

34-i 

13-3 

17-3 

6.1 

n-3 

2.1 

8.4 

3-3 

I6.3 

2-3 

1  26 

35-4 

12.8 

37-9 

!3-i 

2I.O 

5.8 

2-3 

12.5 

3-2 

20.4 

2-5 

27 

39-5 

13.0 

41.7 

12.9 

24.6 

5-5 

18.9 

2-5 

16.5 

3-i 

24.6 

2.7 

28 

43-7 

13.2 

454 

I2.7 

28.2 

5-i 

22.7 

2.6 

20.6 

3-° 

28.7 

2.9  1 

29 

20  47.8 

—134 

o  31.8 

-4-8 

2   26.5 

+    2.8 

4  24.6 

-1-  2.9 

6  32.9 

—  3-i 

30 

5i-9 

13.6 

35-5 

4-5 

30-3 

2.9 

28.7 

2.7 

37-o 

3-3: 

31 

56.0 

13.7 

39-i 

4.2 

32-8 

2.6 

Day. 

July. 

August. 

September. 

October. 

November. 

December. 

0'sB.  4. 

E.T. 

0'SB.A.     E.T. 

0'SR.A. 

E.T. 

0'sB.A. 

E.T. 

0'sB.A. 

E.T. 

0SB.A. 

E.T.  ! 

h      in 

in 

h     m 

in 

h     in 

in 

h     m 

m 

h     m 

771 

h     m 

m 

1 

6  41.1 

-  3-5 

845-9 

—  6.1 

10  42.0 

+   O.I 

12  30.0 

+  10.3 

14  26.3 

+  16.3 

16  30.1 

+  10.7 

2 

45-3 

3-7 

49.8 

6.0 

45  -6 

0.4 

33-6 

10.7 

30.2 

I6.3 

344 

10.4 

3 

494 

3-9 

53-7 

6.0 

49.2 

0.7 

37-3 

10.9 

34-i 

I6.3 

38.7 

IO.O 

4 

53-6 

4.1 

57-5 

5*9 

52-8 

I.I 

40.9 

"•3 

38-1 

16-3 

43-i 

9.6 

5 

57-7 

4-3 

9     i-4 

5-8 

564 

1.4 

44-5 

n.6 

42.0 

I6.3 

474 

9.2 

6 

7    1.8 

44 

5-2 

5-7 

II      O.I 

1.7 

48.2 

11.9 

46.1 

16.2 

51.8 

8.8 

7 

5-9 

4.6 

9.1 

5.6 

3-6 

2.1 

51.8 

12.2 

50.0 

16.2 

56.2 

8-3 

S 

7  10.0 

-4-8 

9  12.9 

—  54 

ii     7-3 

+    2.4 

12  55-5 

+  124 

14  54.0 

+16.1 

17    0.6 

+  7-9 

9 

14.1 

4-9 

16.7 

5-3 

10.8 

2.8 

59-2 

12.7 

58.1 

16.0 

4.9 

74 

jio 

18.2 

5-i 

20.5 

5-2 

14.4 

3-i 

13    2.8 

I3.0 

15      2.1 

15.9 

9-3 

7.0 

11 

22.3 

5-2 

24-3 

5.0 

18.1 

34 

6.5 

13.2 

6.2 

15.8 

13-7 

6-5 

12 

26.3 

5-3 

28.0 

4.8 

21.6 

3-8 

10.2 

J3-5 

10.2 

15-7 

18.1 

6.1 

13 

3°4 

5-5 

31.8 

4-7 

25.2 

4.2 

13-9 

13-7 

14-3 

15.6 

22.6 

5-6 

14 

344 

5.6 

35-6 

4-5 

28.8 

4-5 

I7.6 

14.0 

18.4 

154 

27.0 

5-i 

15 

738.5 

—  $-7 

9  39-3 

-  4-3 

ii  32.4 

+  4-9 

I3    21.4 

+14.2 

15    22.5 

+15-3 

17   314 

+  4-6 

16 

42-5 

s-s 

43-i 

4.1 

36.0 

5-2 

25.1 

14.4 

26.6 

!5-i 

35-8 

4.2 

17 

46.6 

5-9 

46.8 

3-9 

39-6 

5.6 

28.8 

14.6 

30.8 

14.9 

40.3 

3-6 

18 

50.6 

5-9 

50-5 

3-7 

43-2 

5-9 

32.6 

14.8 

34-9 

14.7 

44-7 

3-2 

19 

54.6 

6.0 

54-2 

34 

46.7 

36.3 

15.0 

39-1 

14.4 

49.1 

2-7 

2O 

58.6 

6.1 

58.0 

3-2 

5°4 

6^6 

4O.I 

i5-i 

43-3 

14.2 

53-6 

2.2 

21 

8    2.6 

6.1 

10     1.7 

3-° 

53-9 

7.0 

43-9 

iS-3 

474 

14.0 

58.0 

x-7 

22 

8    6.6 

—   6.2 

10    5-3 

—  2.7 

ii  57-5 

+  7-3 

13  47-7 

+  154 

15  5i-7 

+  I3-7 

18    2.4 

4-  1.2 

23 

10.6 

6.2 

9.1 

2-5 

12      I.I 

7-7 

51-5 

15.6 

55-9 

134 

6.9 

0.7 

i  24 

14-5 

6.2 

12.7 

2.2 

4-7 

8.0 

55-3 

J5-7 

16    o.i 

1Z'1 

11.3 

0.2 

25 

18.5 

6-3 

16.4 

1.9 

8-3 

8.4 

59-i 

15.8 

44 

12.8 

15-8 

!  26 

22.4 

6.2 

20.1 

i-7 

11.9 

8.7 

14    3.0 

8.6 

12.5 

2O.2 

0.8  1 

27 

26.4 

6.3 

237 

1.4 

15.6 

9.0 

6.8 

16.0 

12.9 

12.2 

24.7 

T-3 

;    28 

30.3 

6.2 

27.4 

i.i 

19.4 

94 

10.7 

16.1 

17.2 

1  1.8 

29.1 

1.8 

29 

8  34.2 

—   6.2 

10  31.0  —  0.8 

II    22.8 

+  97 

H  14-5 

+  16.2 

16  21.5 

+11.5 

18  33-5 

—   2.7 

3O 

38-1 

6.2 

34-7        0.5 

26.4 

IO.O 

18.4 

16.2 

25.8 

II.  I 

38.0       -2.8 

31 

42.0 

6.1 

38.3        0.2 

22.3 

16.3 

424       3-3 

180 


TABLE  LX. 


I 

Declination  of  the  Sun. 

1875  :    At  Greenwich  Mean  Noon. 

:Day. 

Jan.    |    Feb. 

Mar. 

Apr. 

May. 

June. 

July.       Aug. 

Sept.        Oct.        XOT.        Dec. 

o 

o 

o 

o 

o 

o 

o 

o 

O              '        O                      <~>                !n                i 

1 

23.0  S. 

17.1  S. 

7.6  S. 

4-5  N- 

i5.oX. 

22.0  X. 

23.  i  X. 

i8.iX. 

8.4  X.    3.18. 

144  S. 

21.88. 

2 
3 

22.9 

22.8 

16.8 
16.6 

7.2 

4-9 
5-3 

'5-3 

15.6 

22.2 
22.3 

23.1 
23.0 

17.8 
17.6 

8.0        3.5 

7-6     i    3-9 

14.7 
15.0 

22.O 
22.1 

4 

22.7 

16.3 

6.5 

5.6 

i5-9 

22.4 

22.9 

17-3 

7-3      i    4-3 

154 

22.2 

5 

22.6 

16.0 

6.1 

6.0 

16.2 

22-5 

22.8 

17.0 

6.9         4.7 

15-7 

22.4 

6 

22-5 

15-6 

5-7 

6.4 

16.5 

22.6 

22.7 

16.8 

6.5         5.1 

16.0 

22-5 

7 

22.4 

15-3 

5-3 

6.8 

1  6.8 

22.7 

22.6 

16.5 

6.1         5-5 

16.3 

22.6 

8 

22.3  S. 

15.08. 

4.98. 

7.2  X. 

17.1  X. 

22.8  X. 

22.5  X. 

16.2  X. 

5.8  X. 

5-88. 

16.68. 

22.78. 

9 

22.1 

14.7 

4-5 

7-5 

17-3 

22.9 

22.4 

15-9 

54 

6.2       16.8 

22.8 

i  1O 

22.0 

14.4 

4-i 

7-9 

17.6 

23.0 

22.3 

15.6 

6.6 

17.1 

22.9 

i  11 

21.8 

14.1 

3-8 

8-3 

17.9       23.1 

22.1 

15-3 

4.6 

7.0 

17.4 

23.0 

i  12 

21.7 

13-7 

34 

8.6 

18.1        23.2 

22.0 

15.0 

4.2 

74 

17.7 

23.I 

13 

21.5 

134 

9.0 

18.4 

23.2 

21.9 

14.7 

3-9 

7-7 

17.9 

232 

i  -*-1* 

21.3 

2.6 

94 

18.6 

23-3 

21.7 

14.4 

3-5 

8.1 

18.2        23.2 

15 

21.  1  S. 

12.7  S. 

2.28. 

9.7x. 

iS.SX. 

23-3  N. 

21.6  X. 

14.  i  X. 

3-iX. 

8.58. 

18.5  S.    23.3  S. 

16 

21.0 

12.4 

1.8 

10.  1 

19.1 

234 

21.4 

13.8 

2-7 

8.8 

18.7 

23-3 

17 

20.8 

12.0 

i-4 

10.4 

19-3 

234 

21.2 

13-5 

2-3 

9.2 

19.0 

234 

18 

20.6 

II.7 

I.O 

10.8 

19-5 

234 

21.  1 

13.2 

1.9 

9-6 

19.2 

23-4 

19 

20.4 

"•3 

0.6 

n.  i 

19.7 

23.4 

2O.9 

12.8 

9-9 

19-5 

234 

1  20 

20.  1 

II.O 

0.28. 

H-5 

20.0 

23-5 

20.7 

12.5 

1.2          10.3 

19.7 

23-4 

i  21 

19.9 

10.6 

0.2  X. 

1  1.8 

2O.2 

23-5 

20.5 

12.2 

0.8       10.7 

19.9 

23-5 

!  22 

19.78. 

10.2  S. 

o.6X. 

12.2  X. 

20.4  X.  i  23.5  X. 

20.3  X. 

H-9X. 

0.4  X.  11.08. 

20.1  S. 

23-5  8. 

1  23 

19.5 

9-9 

I.O 

I2.5 

20.6 

23-4 

20.1 

11.5 

o.olS.   11.4 

20.3 

23-5 

24 

19.2 

9-5 

1.4 

12.8 

20.7 

234 

19-9 

11.2 

0.4 

11.7 

20-5 

234 

i  25 

19.0 

9.1 

1.8 

13.! 

2O-9 

23-4 

197 

10.8 

0.8 

I2.I 

20.7 

234 

26 

18.7 

8.8 

2.2 

*3-5 

21.  1 

23-4 

19-5 

10.5 

1.2 

I2.4 

20.9 

234 

27 

18.5 

8.4 

2.6 

13.8 

21.3 

23-3 

19-3 

10.  1 

1.6 

12.8 

21.  1 

23-3 

28 

18.2 

8.0 

2.9 

14.1 

21.4 

23-3 

I9.O 

9.8 

2.0 

13-1 

21.3 

23-3 

29 

18.0  S. 

3-3  N. 

14.4  X. 

21.6  X. 

23-3  N. 

I8.8X. 

9.4  X. 

2.48. 

1348. 

21.58.     23.28. 

3O 

17.7 

3-7 

14.7 

21.8 

23.2 

18.6 

9.1 

2.7 

13.8 

21.6 

23-2 

31 

17.4 

4.1 

21.9 

18.3 

8.7 

14.1 

23.1 

1876  :    At  Greenwich  Mean  Noon. 

1 

o 

23.0  S. 

0 

17.28. 

7°.3S. 

4.8  X. 

o 

15-3  N. 

22°.  I  X. 

23°.  i  X. 

I7-9N. 

8°.iX. 

3.48. 

i4°.6  S. 

0 

21.98.  j 

2 

23.0 

16.9 

7.0 

5-2 

15.6 

22.3 

23.0 

17.6 

7-7 

3-8 

15.0 

22.1 

3 

22.9 

16.6 

6.6 

5.6 

15-9 

22.4 

22.9 

17.4 

7-3 

4.2 

15.3 

22.2 

4 

22.8 

16.3 

6.2 

5-9 

1O.2 

22.5 

22.8 

17.1 

7.0 

4.6 

15.6 

22.3 

5 

22.7 

16.0 

5-8 

6-3 

16.4 

22.6 

22.7 

16.8 

6.6 

5-° 

15-9 

22-5 

6 

22.5 

'5-7 

54 

6.7 

l6.7 

22.7 

22.6 

1  6.6 

6.2 

54 

16.2 

22.6 

7 

22.4 

154 

5-° 

17.0 

22.8 

22.5 

16.3 

5-9 

5-7 

16.5 

22.7 

8 

22.3  S. 

15.18. 

4.68. 

74  N. 

17.3^. 

22.9  X. 

22.4  X. 

i6.oX. 

5-5  N. 

6.1  S. 

16.88. 

22.8  S. 

9 

22.2 

14.8 

4.2 

7-8 

17.5 

23.0 

22.3 

15-7 

5.1 

6-5 

17.1 

22.9 

10 

22.0 

14-5 

3-8 

8.2 

17.8 

23.1 

22.2 

154 

4-7 

6.9 

17-3 

23.0 

11 

21.9 

14.1 

3-5 

8.6 

18.0 

23.1 

22.0 

IS-1 

4-3 

7-3 

17.6 

23.1 

12 

21.7 

13.8 

3-1 

8.9 

18.3 

23.2 

21.9 

14.8 

4.0 

7-6 

17.9 

23.I 

13 

21-5 

13-5 

2.7 

9-3 

18.5 

23-3 

21.8 

H-5 

3-6 

8.0 

18.2 

23-2 

14 

21.4 

2-3 

9-6   . 

18.8 

23-3 

21.6 

14.2 

3-2 

8.4 

18.4 

23-3 

15 

21.28. 

12.8  S. 

1.98. 

i  o.o  X. 

19.0  X. 

23-3  N. 

21.5  X. 

13.9  X. 

2.8  X. 

8.8  S. 

18.78. 

23-3  s. 

16 

21.0 

12.4 

10.3 

19.2 

234 

21.3 

13.6 

2.4 

9-i 

18.9 

234 

17 

20.8 

I2.I         j     I.I 

10.7 

19-5 

23-4 

21.  1 

13.2 

2.0 

9-5 

19.2 

234 

18 

2O.6 

11.8      1   0.7 

II.O 

19.7 

23-4 

20-9 

12.9 

1.6 

9-9 

19.4 

23.4 

19 

20.4 

11.4 

0.38. 

11.4 

19.9 

234 

20.8' 

12.6 

1.2          10.2 

19.6 

23-4 

20 

2O.2 

II.O 

o.iX. 

11.7 

20.  i 

23-5 

2O.6 

12.3 

0.9 

10.6 

19.9 

23-5 

21 

20.0 

10.7 

0.5 

12.  1 

20.3 

23-5 

20.4 

11.9 

o-5 

10.9 

20.1 

23-5 

22 

19.88. 

10.3  S. 

0.9  X. 

I2.4X. 

20.5  X. 

23.5  X. 

20.2  X. 

n.6X. 

o.iX. 

11.38. 

20.3  S. 

23.5  S. 

23 

!9-5 

10.0' 

1.3 

12.7 

20.7 

234 

20.O 

1  1-3 

0.38.  ii.o 

20.5 

23-4  .  1 

24 

*9-3 

9-6 

1.7 

20.9 

234 

19.8 

10.9 

0.7       I  12.0 

20.7 

234 

25 

19.0 

9.2 

2.1 

134 

21.  1 

23-4 

19-5 

10.6 

I.I        il2.3 

20.9 

23-4 

26 

18.8          8.8 

2-5 

13.7       .'21.2 

234 

19-3 

10.2 

1.5 

12.7 

21.  1 

23-4 

!27 

18.5          8.^ 

2.8 

14.0       i  21-4 

23-3 

I9.I 

9-9 

1.9 

13.0 

21.3 

23-3 

28 

18.3 

8.1 

3-2 

144 

21.6 

23-3 

18.9 

9-5 

2-3 

13-3 

21.4 

23-3 

29 

18.08. 

7-78. 

3-6  X. 

14.7  X. 

2I.7X. 

23.2  X. 

i8.6X. 

9.2  X. 

2.6J3. 

I3-7S. 

21.68. 

23.2  S. 

30 

17.8 

4.0       15.0     1  21.9 

23.2 

18.4 

8.8 

3-° 

14.0 

21.8 

23.1 

31 

17-5       i 

4.4                         |  22.0 

18.1 

8.4 

14-3 

23.1 

TABLE  LX. 


181 


Declination  of  the  Sun. 

1877: 

At  Greenwich.  Mean  Noon. 

Day. 

Jan. 

Feb. 

Mar. 

Apr. 

May. 

June. 

July.   ]    Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1 

23.0  S. 

0 

17.08. 

MS. 

4°.7N. 

o            1      o 
I5.2X.  I22.IX. 

o                  o 

23.  i  N.    I7-9N. 

8.2  X. 

o 
3-3S. 

14.6  S. 

2i°.9S. 

9 

22.9 

16.7 

7.0 

5-i 

15-5 

22.2 

23.0        17.7 

7.8 

3-7 

14.9 

22.O 

3 

22.8 

16.4 

6.7 

I5.8 

22.4 

23.0        17.4 

7-4 

4.1 

15.2 

22.2 

4 

22.7 

16.1 

6-3 

5-8 

16.1 

22.5 

22.9 

17.2 

7-i 

4-5 

15-5 

22.3 

5 

22.6 

15.8 

5-9 

6.2 

16.4 

22.6 

22.8. 

16.9 

6-7 

4-9 

22.4 

6 

22-5 

15.5 

5-5 

6.6 

16.6 

22-7 

22.7 

16.6 

6-3 

5-3 

16.1 

22.6 

7 

22-3 

15.2 

5-i 

7-0 

16.9 

22.8 

22.6 

16.3 

5-9 

5.6 

16.4 

22.7 

8 

22.2  S. 

14.9  8. 

4.78. 

74N- 

17.2  N. 

22.9  N. 

22.  5  X. 

i6.iX. 

5-6X. 

6.08. 

16.78. 

22.8  S. 

9 

22.1 

H-5 

4-3 

7-7 

17-5 

23.0 

22.3 

15.8 

5-2 

6.4 

17.0 

22.9 

to 

21.9 

14.2 

3-9 

8.1 

17.7 

23.1 

22.2 

15-5 

4.8 

6.8 

17-3 

23.0 

11 

21.7 

13-9 

3-5 

8-5 

18.0 

23.1 

22.1 

15.2 

4-4 

7-2 

17-5 

23.1 

12 

21.6 

13.6 

3-2 

8.8 

18.2 

23.2 

21-9 

14.9 

4.0 

7-5 

17.8 

23.1 

1  13 

21.4 

13.2 

2.8 

9.2 

18.5     23.2 

21.8 

14.6 

3-7 

Z'9 

18.1 

23-2 

14 

21.2 

12.9 

2.4 

9.6 

18.7     23.3 

21.6 

H-3 

3-3 

8-3 

18.3 

23-3 

15 

21.  1  S. 

12.58. 

2.08. 

9.9  N. 

19.0  N.   23.3  IST. 

21.5  N. 

i4.oX. 

2.9  X. 

8.78. 

18.6  S. 

23.3S. 

16 

20-9 

12.2 

1.6 

10.3 

19.2 

23-4 

21.3 

13-6 

2-5 

9.0 

18.9 

234 

17 

20.7 

11.8 

1.2 

10.6 

19.4        23.4 

21.2 

13-3 

2.1 

9-4 

19.1 

234 

IS 

20.5 

n-5 

o.S 

II.  0 

19.6        23.4 

21.0 

13.0 

i-7 

9.8 

19-3 

23-4 

19 

20-3 

ii.  i 

0.48. 

"•3 

19.9        23.4 

20.8 

12.7 

i-3 

10.  1 

19.6 

234 

20 

2O.O 

10.8 

o.o 

11.7 

20.1 

23-5 

20.6 

12.3 

I.O 

10.5 

19.8 

23-5 

21 

19.8 

10.4 

0.4  N. 

12.0 

20.3 

23-5 

20.4 

12.0 

0.6 

10.8 

2O.O 

23-5 

22 

19.6  S. 

10.0  S. 

0.8  N. 

I2.3X. 

20.5  N. 

23.5  N. 

20.2  N. 

ii.  7N. 

0.2X. 

11.28. 

20.2  S. 

23-5  s. 

23 

19-3 

9-7 

1.2 

12.7 

20.7 

23-4 

2O.O 

n-3 

0.28. 

n-5 

2O.4 

23-4 

21 

19.1 

o,  3 

1.6 

13.0 

20.8 

23-4 

19.8 

II.O 

0.6 

11.9 

20.6 

234 

25 

18.9 

8.9 

2.0 

13-3 

21.0 

23-4 

19.6 

10.7 

1.0 

12.2 

20.8 

23-4 

26 

18.6 

8.6      i    2.4       13.6 

21.2 

23-4 

194 

10.3 

1.4 

12.6 

21.0 

23-4 

27 

18.4 

8.2 

2.8 

14.0 

21.4         ,23.3 

19.2 

10.0 

1.8 

I2.9 

21.2 

23-3 

28 

iS.i 

7.8 

3-i 

14-3 

21-5           23.3 

l8.9 

9.6 

2.2 

13-3 

21.4 

23-3 

29 

17.88. 

3.5  N.  14.6  N. 

21.  7  N.    23.2  N. 

i8.;X. 

9.2  N. 

2.68. 

13.68. 

21.68. 

23.2  S. 

30 

17-5' 

3-9       14-9 

21.8        23.2 

18.6 

8.9 

2.9 

13-9 

21.7 

23.2 

31 

17-3 

4-3 

22.0 

18.2 

8.5      |              14.2     ! 

23.1 

1878 

At  Greenwich  Mean  Noon. 

1 

23.0  S. 

0 

17.1  S. 

7-5  S- 

4°.6X. 

i5°.iN.   22.  i  N. 

o 

23.  iN. 

iS.oXJ    8.2  X. 

o 
3.28. 

o 

14.5  S. 

21.88. 

2 

22.9 

16.8 

7-i 

5-o 

15-4 

22.2 

23.0 

17.8 

7-9 

3-6 

14.8 

22.0 

3 

22.8 

16.5 

6.8 

5-4 

15.7       22.3 

23.0 

17-5 

7-5 

4.0 

15-1 

22.1 

4 

22.7 

16.2 

6-4 

5.8 

16.0        22.5 

22.9 

17.2 

7-i 

4-4 

15-4 

22.3 

5 

22.6 

i5-9 

6.0 

6.1 

16.3           22.6 

22.8 

17.0    - 

6.8 

4.8 

15-7 

22.4 

6 

22.5 

15-6 

5.6 

6.5 

16.6 

22-7 

22.7 

16.7 

6.4 

5-2 

16.0 

22.5 

7 

22.4 

i$-3 

5-2 

6.9 

16.9 

22.8 

22.6 

16.4 

6.0 

5.6 

16.3 

22.6 

§ 

22.2S. 

14.9  S. 

4.88. 

7-3  N. 

17.1  N. 

22.9  N. 

22.5  N. 

i6.iX. 

5.7x. 

5.98. 

16.68. 

22.88. 

9 

22.1 

14.6 

4.4 

7.6 

17.4 

23.0 

22.4 

15.8 

5-3 

6-3 

16.9 

22.9 

10 

21.9 

H-3 

4.0 

8.0 

17-7 

23.0 

22.2 

15-5 

4-9 

6.7 

17.2 

22.9 

11 

21.8 

14.0 

3-6 

8.4 

17.9 

23.1 

22.1 

15-3 

4-5 

7-i 

17-5 

23-0' 

12 

21.6 

13.6 

3-3 

8.7 

18.2 

23.2 

22.O 

15.0 

4.1 

7-5 

23.1 

13 

21.5 

J3-3 

2.9 

9.1 

18.4 

23.2 

21.8 

14.6 

3-8 

7-8 

18.0 

23.2 

14 

21.3 

13.0 

2-5 

9-5 

18.7 

23-3 

21.7 

H-3 

3-4 

8.2 

18.3 

23.2 

15 

21.18. 

12.6  S. 

2.IS. 

9.8  N. 

18.9  N. 

23.3  N. 

21.5  N. 

14.  i  X. 

3.0  X. 

8.68. 

18.5  S. 

23-3  s. 

16 

20.9 

12.3 

1-7 

10.2 

19.1 

23-4 

21.4 

13-7 

2.6 

8.9 

18.8 

23.3 

17 

20.7 

11.9 

i-3 

I0.5 

19.4 

23-4 

21.2 

13-4 

2.2 

9-3 

19.0 

23-4 

18 

20.5 

n.6 

0.9 

10.9 

19.6 

23-4 

2I.O 

I3-1 

1.8 

9-7 

19-3 

23.4 

19 

20.3 

11.2 

o-5 

II.  2 

I9.« 

23-4 

20.8 

12.8 

1.4 

IO.O 

19-5 

234 

i  20 

20.1 

10.9 

o.i  S. 

11.6 

2O.O 

23-5 

20.7 

12.4 

I.O 

10.4 

19.7 

23-4 

21 

19.9 

10.5 

0.3  N 

11.9 

20.2         |  23.5 

20.5 

I2.I 

0.7 

10.8 

2O.O 

23.5 

22 

19.68. 

10.1  S. 

o.7N 

12.2  X. 

20.4  N. 

23.5  N. 

20.3  N. 

ii.SX. 

0.3  X. 

ii.  18. 

20.2  S. 

23.5  s. 

23 

194 

9.8 

i.i 

12.6 

:  20.6 

23-4 

20.  1 

11.4 

0.18.   11.5 

20.4 

23-4 

24 

I9.2 

9.4 

i-5 

12.9 

20.8      1  23.4 

19.9 

ii.  i 

0.5       i  1.8 

20.6 

23-4 

25 

18.9 

9.0 

1.9 

13.2 

21.0         i23.4 

19.7 

10.7 

0.9          12.2 

20.8 

23-4 

26 

I8.7 

8.7 

2-3 

13-6 

21.2         ^23.4 

19.4 

'10.4 

1-3          12-5 

21.0 

23-4 

27 

l8.4 

8-3 

2-7 

13-9 

21.3    :23.3 

19.2 

IO.O 

1.7       12.8 

21.2 

23-3 

28 

18.2 

7-9 

3-i 

I4.2 

21-5           23.3 

19.0 

9-7 

2.1          13.2 

21.4 

23-3 

29 

17.98. 

34* 

14.5  N 

2  1.  6  N.   23.2  N. 

18.7  N. 

9.3  X. 

2.SS.  '13.58. 

21.58. 

23.2  8. 

3O 

17.6 

3-8 

14.8 

21.8       i  23.2 

18.5 

9.0 

2.Q          13.8 

21.7           23.2 

31 

J7-3 

4.2 

21.9 

18.3 

8.6 

14.2 

23.1 

182 


TABLE  LXI. 


Mean  Places  of  Twenty-five  Fixed  Stars.  1875  :  January  1  ,  Greenwich. 

Common 
Name. 

! 

Right 
Ascen- 
sion. 

Annual 
Difif. 

Decli- 
nation. 

Annual 
Diff. 

Name  of  Con- 
stellation. 

Systematic 
Name  of  Star. 

h     m 

m 

o 

o 

Alpheratz, 

2 

o     1.93 

0.05 

28.40  N 

-{-0.005 

Androm'eda, 

a  Andromed  e. 

Polaris  (P.  Star), 

2 

I   13.00 

35 

88.64  N. 

+   005 

The  Little  Bear, 

a  Ursse  Minoris. 

Acher'nar, 

I 

i  33-°5 

04 

57.878. 

—  005 

Erid'anus, 

a  Eridani, 

Hamel, 

2 

2      0.13' 

06 

22.87  N 

+   005 

The  Ram, 

a  Arietis. 

Aldebar'an, 

I 

4  28.75 

06 

16.26  N 

+   002 

The  Bull, 

a  Tauri. 

Capella, 

5    746 

0.07 

45.87  N 

-j-O.OOI 

The  Charioteer, 

a  Aurigce. 

Rigel, 

5    8.53 

°5 

8.35  s. 

—    001 

Ori'on, 

ft  Orionis. 

Betelguese, 

Var 

5  48.41 

°5 

7.38  N 

4-   ooo 

Ori'on, 

a  Orionis, 

Cano'pus, 

6  21.18 

02 

52.638. 

-j-   ooo 

The  Ship  Argo, 

a  Argus. 

Sir'ius, 

6  39.64 

04 

16.55  S. 

-j-  ooi 

The  Great  Dog, 

a  Canis  Majoris. 

Castor, 

7  26.62 

0.06 

32.I6N 

—  O.OO2 

The  Twins, 

a  Geminorum. 

Pro'cyon, 

7  32-76 

°5 

5.54  N 

—    002 

The  Little  Dog, 

a  Canis  Minoris. 

Pollux, 

7  37-66 

06 

28.33  N 

—    OO2 

The  Twins, 

ft  Geminorum. 

Alpharcl, 

9  21.45 

05 

8.12  S. 

+     004 

The  Hydra, 

a  Hydrce. 

Reg'ulus, 

10    1.71 

05 

12.58  N 

—    005 

The  Lion, 

a  Leonis. 

a1  Crucis, 

12    19.65 

0.05 

62.43  S. 

+0.005 

The  Cross, 

a1  Crucis. 

Spica, 

13  18.61 

°5 

10.51  S. 

-f-     005 

The  Virgin, 

a  Virginis. 

ft  Centauri, 

I3  55-02 

07 

59-77  S. 

-r  005 

The  Centaur, 

ft  Centauri. 

Arctu'rus, 

14    9.96 

°5 

19.83  N 

—  005 

The  Bear-  Watcher, 

a  Bootis. 

a2  Centauri, 

14  31.14 

07 

60.31  S. 

-{-  004 

The  Centaur, 

a2  Centauri. 

Anta'res, 

16  21.75 

0.06 

26.158. 

-{-0.002 

The  Scorpion, 

a  Scorpii. 

Vega, 
Altair, 

1  18  32.71 
19  44.68 

03 
°5 

38.67  N 
8.54  N 

4-   ooi 

-j-  003 

The  Harp, 
The  Eagle, 

a  Lyrre. 
a  Aquilce. 

Fo'malhaut, 

!|22   50.74 

°5 

30.288. 

—  005 

The  Southern  Fish, 

a  Piscis  Australis. 

Mar'kab, 

2      22    58.53 

05 

14-53  N 

.  +  005 

Peg'asus, 

a  Pegasi. 

The  DifF.  in  R.  A.  ia  always  additive.    The  DifF.  in  Dec.  is  additive  or  subtractive 

according  as  the  sign  is  +  or  —  . 

TABLE  LXII. 

Meridian-Passages  of  Twenty-five  Fixed  Stars. 

1875  :   At  Greenwich  Mean  Noon  on  the  First  Day  of  the  Month. 

Name  of 
Star. 

Jan. 

Feb. 

Mar. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

h  m 

h   m 

h  m 

h   m 

h   m 

h  m 

h  m 

h  m 

k  m 

h    m 

h    m 

h    in 

Alpheratz, 

5  i5 

3    3 

I  14 

2320 

21  29 

19  26 

17  22 

15  17 

1321 

ii  33 

937 

733 

Polaris, 

626 

414 

225 

031 

22  40 

2037 

1833 

1628 

1432 

1244 

10  48 

844  | 

Achernar, 

646 

434 

245 

051 

23    o 

2057 

1853 

1648 

1452 

13    4 

ii    8 

9    4  i 

Hamel, 

7i3 

5    i 

312 

i  18 

2327 

21  24 

19  20 

1715 

15  19 

I33i 

"35 

93i  i 

Aldebaran, 

942 

730 

541 

347 

I56 

2353 

2M9 

1944 

1748 

16    o 

14   4 

12     O 

Capella, 

10  21 

8   9 

6  20 

426 

235 

032 

2228 

20  23 

1827 

1639 

1443 

1239 

Rigel, 

1022 

8  10 

621 

427 

236 

033 

22  29 

2024 

1828 

1640 

1444 

I240 

Betelguese, 

II     2 

850 

7    i 

5    7 

316 

I   13 

23    9 

21     4 

19    8 

16  20 

1524 

13  2O 

Canopus, 

H34 

922 

733 

539 

348 

145 

2341 

21  36 

1940 

1752 

1556 

1352 

Sirius, 

"53 

941 

752 

558 

4    7 

2     4 

24   o 

2155 

1959 

18  ii 

1615 

14  II 

Castor, 

12  40 

1028 

839 

645 

454 

251 

047 

22  42 

2046 

1858 

I7     2 

1458 

Procyon, 

12  46 

1034 

845 

651 

5    o 

257 

053 

2248 

2052 

19    4 

17    8 

15    4 

Pollux, 

1251 

1039 

850 

656 

5    5 

3    2 

058 

2253 

2057 

19    9 

1713 

15    9 

Alphard, 

H35 

1223 

1034 

840 

649 

446 

242 

037 

2241 

2053 

i857 

i653 

Regulus, 

1515 

13    3 

ii  14 

9  20 

729 

526 

322 

I   17 

23  21 

21  33 

1937 

1733 

a1  Crucis, 

1733 

15  21 

1332 

ii  38 

947 

744 

540 

335 

139 

235i 

21  55 

I95i 

Spica, 

1832 

16  20 

1431 

1237 

1046 

843 

639 

434 

238 

o  50 

2254 

20  50 

ft  Centauri, 

19    8 

1656 

15    7    13  J3 

II  22 

919 

715 

5  10 

3  14 

i  26 

2330 

21  26 

Arcturus, 

1.923 

17  II 

15  22  |  13  28 

H37 

934 

730 

525 

329 

i  41 

2345 

2I4I 

a'2  Centauri, 

1944 

1732 

1543 

13*49 

1158 

955 

751 

546 

35° 

2     2 

o    6 

22     2 

Antares, 

2135 

1923 

1734 

1540 

1349 

ii  46 

942 

737 

54i 

353 

i  57 

2353 

Vega, 

2346 

21  34 

1945 

1751 

16    o 

1357 

ii  53 

948 

752 

6    4 

4    8 

2     4! 

Altair, 

058 

22  46 

2057 

19   3 

1712 

15    9 

13    5 

II     O 

9  -4 

716 

52J 

316  ! 

Fomalhaut, 

4    4 

1  52 

0    3 

22     9 

20  18 

1815 

16  ii 

14    6 

12  IO 

IO  22 

8  26 

6  22  : 

Markab, 

412 

2     0 

0  II 

22  17 

20  26 

1823 

16  19 

H  14 

12  18 

1030 

834 

630 

TABLE  LXIII. 


183 


Reductions  of  Meridian  Passages  of  the  Fixed  Stars. 

1875  :   At  Greenwich  Mean  Noon  on  each  day  of  the  month. 

Day. 

Jan. 

Feb. 

Mar. 

Apr. 

May. 

June. 

July. 

Aug. 

Sept. 

Oct. 

Nov. 

Dec. 

1 

Oh    Om 

Oh    Om 

oh    om 

Oh    Om 

Oh    Om 

Oh    om 

Oh    Om 

Q\i       Qm 

Oh    Om 

Oh   Om 

Q!I   om 

Oh    Om 

2 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

4 

3 

9 

8 

7 

7 

8 

8 

8 

8 

7 

7 

8 

9 

4 

13 

12 

II 

ii 

II 

12 

12 

12 

II 

II 

12 

!3 

5 

18 

16 

15 

14 

15 

16 

16 

15 

14 

14 

16 

17 

0 

2.2 

20 

19 

18 

19 

21 

21 

19 

18 

18 

20 

22 

7 

26 

24 

22 

22 

23 

25 

25 

23 

22 

22 

24 

26 

8 

o  31 

o  28 

0   26 

o  25 

o  27 

o  29 

o  29 

o  27 

o  25 

o  25 

o  28 

o  30 

9 

35 

32 

3« 

29 

31 

33 

33 

31 

29 

29 

32 

35 

10 

39 

36 

33 

33 

35 

37 

37 

35 

32 

33 

36 

39 

11 

44 

40 

37 

36 

39 

4i 

4i 

38 

36 

36 

40 

^44 

12 

48 

44 

4i 

40 

43 

45 

45 

42 

40 

40 

44 

48 

13 

52 

48 

44 

44 

46 

49 

49 

46 

43 

44 

48 

52 

11 

57 

•     52 

48 

47 

50 

54 

53 

5o 

47 

48 

52. 

57 

15 

i     i 

o  56 

o  52 

o  5i 

o  54 

o  58 

o  57 

o  53 

o  50 

o  5i 

o  56 

i     i 

16 

5 

I      O 

55 

55 

58 

I       2 

i     i 

57 

54 

55 

I      0 

6 

17 

10 

3 

59 

59 

I      2 

6 

5 

i     i 

58 

59 

4 

10 

18 

H 

7 

i     3 

I       2 

6 

10 

9 

5 

i     i 

I      2 

9 

15 

19 

18 

ii 

6 

6 

10 

15 

!3 

8 

5 

6 

13 

19 

2O 

22 

15 

10 

10 

H 

«9 

17 

12 

8 

10 

17 

23 

21 

27 

19 

H 

13 

18 

23 

21 

16 

12 

14 

21 

28 

22 

1    31 

i  23 

i  17 

i  17 

I    22 

i  27 

I    2j 

i  19 

i  16 

i  18 

I    25 

i  32 

23 

35 

26 

21 

21 

26 

3i 

29 

23 

19 

21 

29 

37 

24 

39 

30 

25 

25 

30 

35 

33 

27 

23 

25 

34 

4i 

25 

43 

34 

28 

28 

34 

39 

37 

3i 

26 

29 

38 

46 

26 

48 

38 

32 

32 

38 

43 

4i 

34 

30 

33 

42 

50 

27 

52 

42 

35 

36 

42 

48 

45 

38 

34 

37 

46 

55 

28 

56 

45 

39 

40 

46 

52 

49 

42 

37 

4i 

5i 

59 

29 

2      0 

i  49 

*  43 

i  44 

i  50 

i  56 

i  53 

i  45 

i  41 

i  44 

i  55 

2    3 

3O 

4 

46 

47 

55 

2      0 

57 

48 

44 

48 

59 

8 

31 

8 

50 

59 

2      I 

52 

52 

12 

The  Reduction  is  to  be  subtracted  from  the  Time  in  Table  LXII.    The  result  -will 

be  Apparent  Time  ;  and  this,  although  adapted  to  1875,  will  be  within  2m  for  many 

years. 

184 


TABLE  LXIV. 


Reduction  of  Daily  or  Hourly  changes  in  Right  Ascension. 

Daily 
Change. 

Decimal  Parts  of  the  Hour. 

Hourly 
Change. 

Oh.O 

0".l 

0".2 

Oh.3 

Oh.4 

0".5 

0!l.6 

0'  7 

0".8 

0".9 

lh.O 

m 

m 

m 

in 

in 

m 

m 

m 

in 

m 

m 

m 

i' 

o.o 

O.O 

0.00 

o.oo 

o.oo 

o.oo 

o.oo 

o.oo 

o.oo 

0.00 

o.oo 

o.oo 

o.oo 

1 

o 

oo 

00 

00 

00 

00 

00 

00 

00 

oo 

00 

25 

2 

0 

oo 

oo 

oo 

00 

oo 

00 

01 

OI 

OI 

01 

50 

3 

Q 

00 

00 

00 

oo 

OI 

OI 

OI 

01 

01 

OI 

75 

4 

Q 

00            00 

oo 

-01 

01 

01 

01 

01 

01 

02 

l.OO 

0.5 

0.0 

o.oo      o.oo 

O.OI 

O.OI 

O.OI 

O.OI 

O.OI 

0.02 

0.02 

O.O2 

1.25 

6 

o 

oo         oo 

01 

01 

01 

OI 

02 

02 

02 

02 

50 

7 

Q 

00            01 

01 

ol 

02 

02 

02 

02 

°3 

°3 

75 

8 

o 

OO             OI 

OI 

OI 

02 

02 

02 

°3 

03 

°3 

2.OO 

9 

o 

OO        •    OI 

OI 

01 

02 

02 

°3 

°3 

03 

04 

25 

1.0 

0.0 

O.OO       O  OI        O.OI 

0.02 

O  O2         O.O2 

0.03 

0.03 

0.04 

o  04 

250 

1 

o 

00            01 

01 

02 

02   :      03 

03 

04 

04 

°5 

75 

2 

o 

oo 

01 

01 

02 

02 

03 

03 

04 

04 

05 

3.OO 

3 

o 

OI 

OI 

02 

02 

°3 

03 

04 

04 

°5 

05 

25  j 

4 

o 

01 

OI 

02 

02 

03 

03 

04 

05 

05 

06 

50 

1.5 

o.o 

O.OI 

O.OI 

002 

O.O2 

0.03 

0.04 

0.04 

0.05 

0.06 

0.06 

3.75 

6 

o 

OI 

OI 

02 

°3 

°3 

04 

05 

05 

06 

07 

4.OO 

7 

o    !      or 

OI 

02 

°3 

03 

04 

05 

06 

06 

07 

25  ; 

8 

o          01 

01 

02 

03 

04 

04 

05 

06 

07 

07 

50 

9 

o          01 

02 

02 

03 

04 

°5 

06 

06 

07 

08 

75  | 

2.0 

O.O          O.OI 

o  02 

O.O2 

0.03 

0.04 

0.05 

0.06 

0.07 

0.07 

0.08 

5  OO 

1 

0              01 

02 

03 

03 

04 

05 

06 

07 

08 

09 

25 

2 

O               OI 

02 

°3 

04 

05 

05 

06 

07 

08 

09 

50 

3 

Q 

01 

02 

03 

04 

05 

06         07 

08 

09 

10 

75 

4 

o 

OI 

02 

03 

04 

05 

06   ;      07 

08 

09 

IO 

6.OO 

2.5 

o.o    !  o.oi      0.02 

0.03 

0.04 

0.05 

o  06 

0.07 

0.08 

0.09 

O.IO 

6.25 

6 

O               OI 

02 

03 

04 

05 

06 

08 

09 

IO 

II 

50 

7 

O             OI    !        O2 

03 

04 

06 

07 

08 

09 

10 

II 

75   | 

8 

0              01    \         02 

03 

05 

06 

°7 

08 

09 

10 

12 

7.OO 

9 

0 

01 

02 

04 

05 

06 

07 

09 

IO 

II 

12 

25 

3.0 

0  0 

O.OI 

O.O2 

o  04 

005 

0.06 

0.07 

0.09 

0  10 

0  II 

0.  12 

7.5O 

1 

o 

01 

03 

04 

05 

06 

08 

09 

10 

12 

13 

75   ! 

2 

o 

OI 

°3 

04 

05 

07 

08 

09 

II 

12 

'3 

§.OO 

3 

o 

01 

03 

04 

05 

07 

08 

IO 

II 

12 

14 

25 

4 

o 

01 

03 

04 

06 

07 

08 

10 

II 

13 

14 

50 

3.5 

0.0 

O.OI 

0.03 

0.04 

0.06 

0.07 

0.09 

O.IO 

0.12 

0.13 

0.15 

8.75 

6 

0 

01 

03 

04 

06 

07 

09 

10 

12 

13 

15 

9.OO  j 

7 

o 

02 

°3 

05 

06 

08 

09 

II 

12 

14 

15 

25 

8 

o 

02 

°3 

05 

06 

08 

09 

II 

13 

14 

16- 

50 

9 

0 

02 

03 

05 

06 

08 

IO 

II 

13 

J5 

16 

75 

4.0 

00 

0.02 

0.03 

0.05 

007 

0.08 

O.IO 

0.12 

013 

0.15 

0.17 

10.00 

1* 

o 

02 

°3 

05 

07 

09 

10 

12 

14 

15 

17 

25 

2 

0 

02 

°3 

05 

07 

09 

10 

12 

14 

16 

17 

5O 

3 

o 

02 

04 

05 

07 

09 

II 

13 

14 

16 

18 

75  ; 

4 

o 

02 

04 

05 

07 

09 

II 

13 

15 

16 

18 

11.00 

4.5 

o.o 

O.O2 

0.04 

0.06 

0.08 

0.09 

0.  II 

0.13 

0.15 

0.17 

0.19 

1125 

6 

0 

02 

04 

06 

08 

09 

II 

13 

15 

17 

*9 

50  ; 

7 

o 

02 

04 

06 

08 

10 

12 

14 

16 

18 

20 

75 

8 

o 

02 

04 

'06 

08 

10 

12 

14 

16 

18 

20 

12.00 

9 

0 

02 

04 

06 

08 

10 

12 

14 

16 

18 

20 

25 

5.0 

0.0 

O.O2 

o  04 

0.06 

0.08 

0  10 

0.12 

0.15 

0.17 

0.19 

O.2I 

12.50 

1 

o 

02 

04 

06 

08 

II 

J3 

15 

17 

19 

21 

75  i 

2 

o 

02 

04 

06 

09 

II 

I3 

15 

17 

19 

22 

13.OO 

3 

0 

02 

04 

07 

09 

II 

'3 

15 

18 

20 

22 

25 

4 

o 

02 

04 

07 

09 

II 

13 

16 

18 

20 

22 

50 

5.5 

0.0 

0.02 

0.05 

0.07 

0.09 

0.  II 

0.14 

0.16 

0.18 

0.21 

0.23 

13.75 

6 

o 

02 

°5 

07 

09 

12 

14 

16 

19 

21 

23 

14.OO 

7 

o 

02 

°5 

07 

09 

12 

14 

17 

19 

21 

24 

25 

8 

0 

02 

°5 

07 

IO 

12 

14 

17 

!9 

22 

24 

5O 

9 

0 

02 

°5 

07 

•10 

12 

14 

17 

20 

22 

25 

75 

6.0 

0.0 

002 

005 

0.07 

O.IO 

0.13 

0.15 

0.17 

0.20 

0.22 

0.25 

15.00 

TABLE  LXIV. 


185 


Reduction  of  Daily  or  Hourly  Changes  in  Right  Ascension. 

Daily 
,  Change. 

Hours  of  the  Day. 

Hourly 
Change. 

2*0 

4".0 

6b.O 

8h.O 

10bO 

12h.O 

14h.O 

16b.O 

18M> 

20h.O 

22h.O 

m 

in 

m 

m 

m, 

m 

m 

771 

m 

m 

m 

m 

s 

o.o 

O.OO 

o.oo 

0.00 

O.OO 

0.00 

0.00 

0.00 

0.00 

O.OO 

0.00 

O.OO 

o.oo 

1 

01 

01 

O2 

03 

04 

05 

06 

06 

07 

08 

09 

25 

02 

°3 

05 

07 

08 

IO 

12 

13 

15 

16 

18 

50 

3 

02 

°5 

07 

IO 

12 

15 

17 

20 

22 

25 

27 

75 

4 

03 

07 

IO 

'3 

17 

20 

23 

27 

30 

33 

37 

1.00 

0.5 

0.04 

0.08 

0.12 

0.17 

O.2I 

0.25 

0.29 

°-33 

0.38 

0.42 

0.46 

1.25 

6 

05 

10 

15 

20 

25 

30 

35 

40 

45 

50 

55 

5O 

7 

06 

12 

17 

23 

29 

35 

41 

46 

52 

S8 

64 

75 

8 

07 

13 

20 

27 

33 

40 

46 

53 

60 

67 

73 

2.00 

9 

07 

15 

22 

3° 

37 

45 

52 

60 

67 

75 

82. 

25 

1.0 

0.08 

0.17 

0.25 

o-33 

0.42 

0.50 

0.58 

0.67 

0.75 

'  0.83 

0.92 

2.5O 

1 

09 

18 

27 

37 

46 

55 

64 

73 

82 

92 

I.OI 

75 

2 

10 

20 

3° 

40 

60 

70 

80 

90 

I.OO 

IO 

3.00 

3 

II 

22 

32 

43 

54 

65 

76 

86 

97 

08 

19 

25 

4 

12 

23 

35 

47 

58 

70 

81 

93 

1.05 

17 

28 

50 

1.5 

0.12 

0.25 

0.37 

0.50 

0.62 

0.75 

0.87 

I.OO 

1.  12 

1.25 

1.37 

3.75 

6 

13 

27 

40 

53 

67 

80 

93 

07 

20 

33 

47 

4.OO 

7 

H 

28 

42 

57 

71 

85 

99 

13 

27 

42 

56 

25 

$ 

30 

45 

60 

75 

90 

1.05 

20 

35 

50 

65 

50 

9 

16 

S2 

47 

63 

79 

95 

ii 

26 

42 

58 

74 

75 

2.0 

0.17 

o-33 

0.50 

0.67 

0.83 

I.OO 

1.17 

i-33 

1.50 

1.67 

1.83 

5.OO 

1 

17 

35 

52 

70 

87 

05 

22 

40 

57 

75 

92 

25 

2 

18 

37 

55 

73 

92 

10 

28 

47 

65 

83 

2.02 

5O 

3    • 

19 

38 

58 

96 

15 

34 

92 

II 

75 

4 

20 

40 

60 

80 

I.OO 

20 

40 

60 

80 

2.00 

20 

6.00 

2.5 

0.21 

0.42 

0.62 

0.83 

1.04 

1.25 

1.46 

1.66 

1.87 

2.08 

2.29 

6.25 

6 

22 

43 

65 

87 

08 

30 

73 

95 

17 

38 

50 

7 
8 

22 
23 

45 

70 

90 
93 

12 
17 

35 
40 

1 

80 

87 

2.02 
IO 

25 

33 

47 
57 

75 
7.OO 

9 

24 

48' 

72 

97 

21 

45 

69 

94 

17 

42 

66 

25 

3.0 

0.25 

0.50 

o-75 

I.OO 

1.25 

1.50 

i-75 

2.00 

2.2S 

2.50 

2-75 

7.5O 

1 

26 

52 

77 

03 

29 

55 

81 

06 

32 

84 

75 

2 

27 

53 

80 

07 

33 

60 

86 

13 

40 

67 

93 

8.00 

3 

27 

55 

82 

10 

37 

65 

92 

20 

47 

75 

3.02 

25 

4 

28 

57 

85 

13 

•  42 

70 

98 

27 

55 

83 

12 

50 

3.5 

O.29 

0.58 

0.87 

1.  17 

1.46 

1.75 

2.04 

2-33 

2.62 

2.92 

3.2I 

8.75 

6 

30 

60 

9° 

20 

80 

IO 

40 

70 

3.00 

3° 

9.00 

7 

31 

62 

92 

23 

54 

85 

16 

46 

77 

08 

'IQ 

25 

8 

32 

63 

95 

27 

58 

90 

21 

53 

85 

17 

48 

50 

9 

32 

65 

97 

30 

62 

95 

27 

60 

92 

25 

57 

75 

4.0 

0.33 

0.67 

I.OO 

i-33 

1.67 

2.00 

2-33 

2.67 

3.00 

3-33 

3-67 

I  O.OO 

1 

34 

68 

02 

37 

71 

05 

39 

73 

07 

42 

76 

25 

2 

35 

70 

05 

40 

75 

10 

45 

80 

15 

50 

85 

5O 

3 

36 

72 

07 

43 

79 

15 

51 

86 

22 

58 

94 

75 

4 

37 

73 

IO 

47 

83 

20 

56 

93 

30 

67 

4-03 

11.00 

4.5 

0-37 

0-75 

1.  12 

1.50 

1.87 

2.25 

2.62 

3.00 

3-37 

3-75 

4.12 

11.25 

6 

38 

77 

15 

53 

92 

30 

68 

07 

45 

83 

22 

50 

7 

39 
40 

11 

17 

20 

11 

96 

2.00 

35 
40 

74 
80 

20 

g 

92 
4.00 

31 

40 

75 
12.00 

9 

82 

22 

63 

04 

45 

86 

26 

67 

08 

49 

25 

5.0 

0.42 

0.83 

1.25 

1.67 

2.08 

2.50 

2.91 

3-33 

3-75 

4.17 

4.58 

12.5O 

1 

42 

85 

27 

70 

12 

55 

97 

40 

82 

25 

67 

75 

2 

43 

87 

30 

73 

17 

60 

3-03 

47 

90 

33 

Z7 

13.00 

3 

44 

88 

32 

21 

65 

09 

53 

97 

42 

86 

25 

4 

45 

90 

35 

80 

25 

70 

15 

60 

4-05 

5° 

95 

50 

5.5 
6 

0.46 
47 

0.92 
93 

40 

1.83 
87 

2.29 

33 

285- 

3.21 

26 

3.66 
73 

4.12 
20 

4.58 
67 

5-04 
13 

13.75 
14.00 

7 

47 

95 

42 

90 

37 

85 

32 

80 

27 

75 

22 

25 

8 

48 

97 

45 

.  93 

42 

90 

38 

87 

35 

83 

32 

50 

9 

49 

98 

47 

97 

46 

95 

44 

93 

42 

92 

41 

75 

6.0 

0.50 

I.OO 

1.50 

2.OO 

2.50 

3.00 

3-5° 

4.00  . 

4-50 

5.00 

5-50 

15.00 

186 


TABLE  LXV. 


Reduction  of  Hourly  Changes  in  the  Moon's  Right  Ascension. 

Parts 

Change  in  One  Hour. 

Parts 

Hour. 

11U.O 

lm.l     lra.2 

1^.3 

lm.4 

lm.5 

lm.6 

1-.7 

1-8 

I111.  9 

20 

of  the 
Hour. 

m 

m 

m   . 

m 

m 

m 

m 

M 

m 

tn 

in, 

m 

m 

o 

O.OO 

O.OO 

O.OO 

O.OO 

O.OO 

O.OO 

O.OO 

0.00 

O.OO 

0.00 

O.OO 

0 

1 

02 

02 

02 

02 

02 

02 

03 

°3 

03 

03 

°3 

1 

2 

03 

04 

04 

04 

05 

05 

°5 

06 

06 

06 

07 

2 

3 

°5 

05 

06 

06 

07 

07 

08 

08 

09 

09 

10 

3 

4 

07 

07 

08 

09 

09 

10 

ii 

ii 

12 

13 

13 

4 

5 

0.08 

0.09 

0.10 

O.I  I 

O.I2 

O.I2 

0.13 

0.14 

0.15 

0.16 

0.17 

5 

6 

10 

II 

12 

13 

14 

15 

16 

17 

18 

19 

20 

6 

7 

12 

13 

14 

15 

16 

17 

19 

20 

21 

22 

23 

7 

8 

13 

15 

16 

17 

19 

20 

21 

23 

24 

25 

•27 

8 

9 

15 

16 

18 

19 

21 

22 

24 

25- 

27 

28 

3° 

9 

10 

O.I7 

0.18 

"O.2O 

O.22 

0.23 

O.25 

0.27 

0.28 

0.30 

0.32 

°-33 

10 

11 

18 

20 

22 

24 

26 

27 

29 

31 

33 

35 

37 

11 

12 
13 

20 

22 

22 

26 

26 
28 

,     28 
30 

30 
32 

32 

35 

34 
37 

36 
39 

38 

40 
43 

12 
13 

14 

23 

26 

28 

30 

33 

35 

37 

40 

42 

44           47 

14 

15 

0.25 

•  0.27 

0.30 

0.32 

°-35 

o-37 

0.40 

0.42 

0-45 

0.47 

0.50 

15 

16 

27 

29 

32 

35 

37    l      40 

43 

45 

48 

5i 

53 

16 

17 

28 

31 

34 

37 

40    i      42 

45 

48 

51 

54 

57 

17 

18 

30 

33 

36 

39 

42 

45 

48 

51 

54 

57 

60 

18 

19 

32 

35 

38 

44 

47" 

51 

54 

57 

60 

63 

19 

20 

0-33 

0-37 

0.40 

°-43 

0.47 

0.50 

°-53 

°-57 

0.60 

0.63 

0.67 

2O 

21 

35 

38 

42 

45 

49  . 

52 

56 

59 

63 

66 

70 

21 

22 

37 

40 

44 

48 

51 

55 

59 

62 

66 

7° 

73 

22 

23 
24 

38 
40 

42 
44 

46 
48 

50 

52 

54 
56 

57 
60 

61 

64 

a 

69 

72 

77 
80- 

23 
24 

25 

0.42 

0.46 

0.50 

°-54 

0.58 

0.62 

0.67 

0.71 

0-75 

0.79 

0.83 

25 

26 

43 

48 

52 

56 

61 

65 

69 

74 

78 

82 

8? 

26 

27 

45 

49 

54 

58 

63 

67 

72 

76 

81 

85 

90 

27 

28 

47 

51 

56 

61 

66 

70 

75 

79 

84 

89 

93 

28 

29 

48 

53 

58 

63 

68 

72 

77 

82 

87 

92 

97 

29 

30 

0.50 

°-55 

0.60 

0.65 

0.70 

o-75 

0.80 

0.85 

0.90 

°-95 

I.OO 

3O 

31 

52 

57 

62 

67 

72 

77 

83 

88 

93 

98 

°3 

31 

32 
33 

53 
55 

8 

64 
66 

69 

75 
77 

80 
82 

85 
88 

93 

96 
99 

I.OI 

04 

07 

IO 

32 

33 

34 

57 

62 

68 

74 

79 

85 

91 

96 

1.02 

08 

34 

35 

0.58 

0.64 

0.70 

0.76 

0.82 

0.87 

°-93 

0.99 

1.05 

i.  ii 

1.17 

35 

36 

60 

66 

72 

78 

84 

9° 

96 

1.02 

08 

'4 

20 

36 

37 

62 

68 

74 

80 

86 

92 

99 

°5 

ii 

17 

23 

37 

38 

63 

70 

76 

82 

89 

95 

I.OI 

08 

•    H 

20 

27 

38 

39 

65 

78 

84 

97 

04 

10 

17 

23 

3° 

39 

40 

0.67 

o-73 

0.80 

0.87 

0-93 

I.OO 

1.07 

1.13 

1.20 

1.27 

r-33 

40 

41 

68 

75 

82 

89 

96 

02 

09 

16 

23 

3° 

37 

41 

42 

70 

77 

84 

98 

05 

12 

I9 

26 

33 

40 

42 

43 

72 

86 

93 

I  00 

07 

15 

•22 

29 

36 

43 

43 

44 

73 

81 

88 

95 

03 

10 

17 

25 

32 

39 

47 

44 

45 

0-75 

0.82 

0.90 

0.97 

1.05 

1.  12 

1.20 

1.27 

i-35 

1.42 

1.50 

45 

46 

77 

84 

92 

I.OO 

07 

15 

23 

30 

38 

46 

53 

46 

47 

78 

86 

94 

02 

IO 

17 

25 

33 

41 

49 

57 

47 

48 

80 

88 

96 

04 

12 

20 

28 

44 

52 

60 

48 

49 

82 

90 

98 

06 

14 

22 

'31 

39 

47 

55 

63 

49 

50 

083 

0.92 

I.OO 

1.  08 

I.I7 

1.25 

i-33 

1.42 

1.50 

1.58 

!.67 

5O 

51 

85 

93 

02 

.    10 

19 

27 

36 

44 

53 

61 

70 

51 

52 

53 

87 
88 

95 

97 

°4 

06 

13 
15 

21 

24 

30 
32 

39 

47 
5° 

56 

11 

73 

77 

52 
53    ! 

54 

90 

99 

1.  08 

17 

26 

35 

44 

53 

62 

71 

80 

54    ! 

55 

0.92 

I.OI 

I.IO 

I.I9 

1.28 

i-37 

1.47 

1.56 

1.65 

i-74, 

1.83 

55    i 

56 

93 

°3 

12 

21 

31 

40 

49 

59 

68 

87 

56 

57 

95 

14 

23 

33 

42 

52 

61 

71 

80 

9° 

57 

58 

97 

06 

16 

26 

45 

55 

64 

74 

84 

93 

58 

59 

98 

08 

18 

28 

38 

47 

57 

67 

77 

87 

97 

59 

60 

I.OO          I.IO 

1.20 

I.30 

1.40 

1.50 

i.  60 

1.70 

1.80 

1.90 

2.00 

6O    i 

I'.OOil'.lO 

J«.20 

1*.30 

1--.40    1  -.50 

1*.60 

18.70 

1".80 

1".9O 

2*.00 

L 

Change  in  One  Minute. 

i 

TABLE  LXV.                                 187 

Reduction  of  Hourly  Changes  in  the  Moon's  Right  Ascension^ 

Parts 

**.£  -4-1-*  ^ 

Change  in  One  Hour. 

Parts 

—  f  .  r.  _ 

ot  tne 
Hour. 

2m.O 

2in.l 

2m.2 

2m.3 

2m-4 

2ra.5 

2m.6 

2m.7 

2m.8 

2m.9 

3m.O 

pi  tne 
Hour. 

in 

m 

in 

m 

m 

m 

m 

m 

m 

m 

m 

in 

m 

O 

0.00 

0.00 

O.OO 

0.00 

0.00 

O.OO 

0.00 

0.00 

O.OO 

O.OO 

O.OO 

O 

1 

03 

03           -04 

04 

04 

04 

04 

04 

05 

05 

°5 

1 

2 

07 

07      |       07 

08 

08 

08 

09 

09 

09 

IO 

10 

2 

3 

10 

10 

II 

II 

12 

12 

13 

13 

14 

14 

15 

3 

4 

13 

H 

15 

15 

16 

17 

17 

18 

19 

19 

20 

4 

5 

0.17 

0.17 

0.18 

0.19 

O.2O 

0.21 

0.22 

0.22 

0.23 

0.24 

0.25 

5 

6 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

30 

6 

7 

23 

24 

26 

27 

28 

29 

30 

31 

33 

34 

35 

7 

8 

•27 

28 

29 

31 

32 

33 

35 

36 

37 

39 

40 

8 

9 

30 

31 

33 

34 

36 

37 

39 

40 

42 

43 

45 

9 

10 

0-33 

0-35 

0-37 

0.38 

0.40 

0.42 

0-43 

o-45 

0.47 

0.48 

0.50 

10 

11 

37 

38 

40 

42 

44 

46 

48 

49 

51 

53 

55 

11 

12 

40 

42 

44 

46 

48 

50 

52 

54 

56 

58 

66 

12 

13 

43 

45 

48 

50 

52 

54 

56 

58 

61 

63 

65 

13 

i    14 

47 

49 

54 

56 

58 

61 

63 

65 

68 

70 

14 

15 

0.50 

0.52 

o-55 

0-57 

0.60 

0.62 

0.65 

0.67 

0.70 

0.72    ' 

0-75 

15 

16 

53 

56 

59 

61 

64 

67 

69 

72 

75 

77 

86 

16 

17 

57 

59 

62 

65 

68 

74 

76 

79 

82 

85 

17 

18 
19 

60 
63 

6d 

66 
70 

69 
73 

72 
76 

75 
79 

78 
82 

81 

85 

84 
89 

87 
92 

90 
95 

18 
19 

2O 

0.67 

0.70 

o-73 

0.77 

0.80 

0.83 

0.87 

0.90 

0.93 

0.97 

I.OO 

20 

21 

70 

73 

77 

80 

84 

87 

91 

94 

98 

I.OI 

°5 

21 

22 

73 

77 

Si 

84 

88 

92 

95 

99 

1.03 

06 

IO 

22 

23 

77 

80 

84 

88 

92 

96 

I.OO 

1.03 

07 

ii 

15 

23 

24 

80 

84 

88 

92 

96 

I.OO 

04 

08 

12 

16 

20 

24 

25 

0.83 

0.87 

0.92 

0.96 

I.OO 

1.04 

1.08 

1.  12 

I.I7 

1.  21 

1.25 

.25 

26 

87 

91 

95 

I.OO 

04 

08 

13 

17 

21 

26 

3° 

26 

27 

90 

94 

99 

03 

08 

12 

17 

21 

26 

3° 

35 

27 

28 

93 

98 

1.03 

07 

12 

17 

21 

26 

31 

35 

40 

28 

29 

97 

I.OI 

06 

ii 

16 

21 

26 

30 

35 

40 

45 

29 

3O 

I.OO 

1.05 

I.IO 

1.15 

i.  20 

1.25 

I.30 

'•35 

1.40 

1-45 

1.50 

'  30 

31 

03 

08 

14 

19 

24 

29 

34 

39 

45 

31 

32 

07 

12 

17 

23 

28 

33 

39 

44 

49 

55 

60 

32 

33 

IO 

I5 

21 

26 

32 

37 

43 

48 

54 

59 

65 

33 

34 

13 

19 

25 

3° 

36 

42 

47 

53 

59 

64 

70 

34 

35 

1.17 

1.22 

1.28 

i-34 

1.40 

1.46 

1.52 

I-57 

1.63 

1.69 

J-75 

35 

36 
37 

20 
23 

26 
29 

5 

38 

42 

44 
48 

50 
54 

I 

62 
66 

68 

73 

74 
79 

80 
85 

36 

37 

38 

27 

33 

39 

46 

52 

58 

65 

71 

77 

84 

90 

38 

39 

3° 

36 

43 

49 

56 

62 

69 

75 

82 

88 

95 

39 

40 

i-33 

1.40 

1.47 

i-53 

1.60 

1.67 

i-73 

i.  80 

1.87 

i-93 

2.OO 

40 

41 

37 

43 

5° 

57 

64 

71 

84 

98 

°5 

41 

42 

40 

47 

54 

61 

68 

75 

82 

89 

96 

2-03 

IO 

42 

43 

43 

50 

58 

65 

72 

79 

86 

93 

2.OI 

08 

15 

43 

44 

47 

54 

61 

69 

76 

.83 

91 

98 

05 

13 

20 

44 

45 

1.50 

1.57 

1.65 

1.72 

i.8o 

1.87 

i-95 

2.02 

2.IO 

2.17 

2.25 

45 

46 

53 

61 

69 

76 

84 

92 

99 

07 

15 

22 

3° 

46 

47 

57 

64 

72 

80 

88 

96 

2.04 

II 

19 

27 

35 

47    I 

48 

60 

68 

76 

84 

92 

2.OO 

08 

16 

24 

32 

40 

48    ! 

49 

63 

71 

80 

88 

96 

04              12 

20 

29 

37 

45 

49 

50 

1.67 

r-75 

1.83 

1.92 

2.00 

2.08 

2.17 

2-25 

2-33 

2.42 

2.50 

50 

51 

70 

78 

95 

04 

12 

21 

29 

'      38 

46 

55 

51 

52 

73 

82 

99 

08 

17 

25 

34 

43 

5i 

60 

52    ! 

53 

77 

85 

94 

2.03 

12 

21 

3° 

38 

47 

56 

65 

53    j 

54 

80 

89 

98 

07 

16 

25 

34 

43 

52 

61 

70 

54    i 

55 

1.83 

1.92 

2.02 

2.  II 

2.20 

2.29 

2.38 

2.47 

2-57 

2.66 

2-75 

55    i 

56 

87 

96 

05 

15 

24 

33 

43 

52 

61 

71 

80 

56    I 

57 

90 

99 

09 

18 

28 

37 

47 

56 

66 

75 

85 

57 

58 

93 

2.03 

13 

22 

32 

42 

51 

61 

71 

80 

90 

58 

59 

97 

06 

16 

26 

36 

46 

56 

65 

75 

85 

95 

59 

60 

2.OO 

2.IO 

2.20 

2.30 

2.40 

2.50 

2.60 

2.70 

2.80 

2.90 

3.00 

60 

2s.OO 

2s.  10 

2s.  20 

2s.  3O 

2S.4O 

2s.  5O 

28.60 

2s.  70 

2s.  80 

28.9O 

3S.OO 

| 

Change  in  One  Minute. 

188 


TABLE  LXVI. 


Reduction  of  the  Mean  Sun's  Right  Ascension. 

Tenths  of  the  Hour. 

• 

Time. 

Arc. 

| 

Oh.O 

Ob.l 

Oh.2 

Oh.3 

Oh.4 

Ob.5 

00 

oh.7 

Oh.8 

Oh.9 

ft 
0 

m 
0.00 

in 
O.O2 

m 
0.03 

m 
0.05 

m 
0.07 

m 
O.o8 

m 
O.IO 

m 

O.I  I 

m 
0.13 

m 

0.15 

o 

O 

1 

16 

18 

20 

21 

23 

25 

26 

28 

29 

31 

ir> 

2 

33 

34 

36 

38 

39 

41 

43 

44 

46 

47 

30 

3 

49 

5i 

53 

54 

56 

57 

59 

61 

62 

64 

45 

4 

66 

67 

69 

7i 

72 

74 

75 

77 

79 

80 

60 

5 

0.82 

0.84 

0.85 

0.87 

0.89 

0.90 

0.92 

0.94 

0-95 

0.97 

75 

6 

99 

1.  00 

1.02 

1.03 

1.05 

1.07 

i.  08 

1.  10 

1.  12 

"3 

90 

r 

i-'5 

17 

18 

20 

22 

23 

25 

26 

28 

30 

1O5 

§ 

31 

33 

35 

36 

38 

40 

41 

43 

44 

46 

12O 

9 

48 

49 

5i 

53 

54 

56 

58 

59 

61 

63 

135 

1O 

1.64 

1.66 

1.68 

1.69 

1.71 

1.72 

1.74 

1.76 

1.77 

1.79 

150 

11 

81 

82 

84 

86 

87 

89 

90 

92 

94 

95 

165 

12 

97 

99 

2.OO 

2.O2 

2.04 

2.05 

2.07 

2.09 

2.10 

2.12 

1§0 

13 

2.13 

2.15 

17 

18 

20 

22 

23 

25 

27' 

28 

195 

14 

30 

32 

33 

35 

37 

38 

39 

41 

43 

45 

210 

15 

2.46 

2.48 

2.50 

2.51 

2-53 

2-55 

2.56 

2.58 

2-59 

2.61 

225 

16 

63 

64 

66 

68 

69 

7i 

73 

74 

76 

88 

240 

17 

79 

81 

83 

84 

86 

87 

89 

9i 

92 

94 

255 

18 

96 

97 

99 

3.01 

3.02 

3-04 

3-05 

3-07 

3-°9 

3.10 

27O 

19 

3.12 

3-'4 

3-15 

17 

19 

20 

.22 

24 

25 

27 

285 

20  , 

3.28 

3-3° 

3-32 

3-33 

3-35 

3-37 

3-38 

3-40 

342 

3-43 

300 

21 

45 

47 

48 

5o 

52 

53 

55 

56 

58 

60 

315 

22 

61 

63 

65 

66 

68 

70 

7i 

73 

74 

76 

330 

23 

78 

79 

81 

83 

84 

86 

88 

89 

91 

93 

345 

o°.o 

1°.5 

3°.0 

4°.5 

6°.0 

7°.5 

9°.0 

10°.  5 

12°.0 

13°.5 

Tenths  of  the  Hour  in  Arc. 

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